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RAILROAD CONSTRUCTION. 



THEORY AND PRACTICE. 



A TEXT-BOOK FOB THE USE OF STUDENTS IN 
COLLEGES AND TECHNICAL SCHOOLS. 



BY / 

/ 

WALTER LOEIXG WEBB, C.E., 

Associate Membey American Society of Civil Engineers: 

Assistant Professor of Ciuii Engineering in 

the University of Pennsylvania ; 

etc. 



FIRST EDITION. 
FIRST THOUSAND. 



KEW YORK: 

JOHN WTLEY & SONS. 
London : CITAPNfAN <fe HALL, Limited. 
1900. 



8£C0ND COPY, 



TWO COPIES RECEIVED, 

L Ibrary of COR^ret% 
Office of tli« 

APR 3 1900 

Ke^^Uter of Copyrlfhftk 



1-^ 



■ ^\ 



.61540 

Copyright, 1899, 

BY 

WALTER LORING WEBB. 



/ 




ROBERT DRUMMOND, PRINTKR, NEW YORE. 



PREFACE. 



The preparation of this book was begun several years ago, 
when much of the subject-matter treated was not to be found in 
print, or was scattered through many books and pamphlets, and 
was hence unavailable for student use. Portions of the book 
have already been printed by the mimeograpli process or have 
been used as lecture-notes, and hence have been subjected to 
the retining process of classroom use. 

The author would call special attention to the following 

features : 

a. Transition curves ; the multiform-compound-curve method 
is used, which has been followed by many railroads in this country ; 
the particular curves here developed have the great advantage of 
beino- exceedingly simple, and although the method is not theu- 
retically exact, it is demonstrable that the differences are so 
small that they may safely be neglected. 

h. A system of earthwork computations by means of a slide- 
rule (which accompanies the volume) which enables one to com- 
pute readily the volume of the most complicated earthwork 
forms with an accuracy only limited by the precision of the 
cross-sectioning. 

c. The ^' mass curve " in earthwork; the theory and use of 
this very valuable process. 

d. Tables I, II, III, and lY have been computed ah novo. 
Tables I and II were checked (after computation) with other 
tables, whicli are ofcnerallv considered as standard, and all dis- 
crepancies were further examined. They are believed to be 
perfect. 



Ill 



IV PREFACE. 

e. Tables Y, YI, YII, and IX Lave been borrowed, by per- 
mission, from "Ludlow's Mathematical Tables." It is believed 
that five- place tables give as accurate results as actual field prac- 
tice requires. Tables YIII and X have been compiled to con- 
form with Ludlow's system. 

The author wishes to acknowledge his indebtedness to Mr. 
Chas. A. Sims, civil engineer and railroaJ contractor, for read- 
ing and revising the portions relating to the cost of earthwork. 

Since the book is written primarily for students of railroad 
engineering in technical institutions, the author has assumed the 
usual previous preparation in algebra, geometry, and trio-o- 
nometry. 

Walter Loring Webb. 
University of Pennsylvania, 
Philadelphia, 
Jan. 1, 1900. 



TABLE OF CONTENTS. 



CHAPTER I. 

RAILROAD SURVEYS. 

PAGE 

Reconnoissance 1 

1. Character of a reconnoissance survey. 2. Selection of a general 
route. 3. Valley route. 4. Cross-country route. 5. Mountain route. 
6. Existing maps. 7. Determination of relative elevations. 8. Hori- 
zontal measurements, bearings, etc. 9. Importance of a good 
reconnoissance. 

Preliminary surveys 8 

10. Character of survey. 11. Cross-section method. 12. Cross- 
sectioning. 13. Stadia method. 14. "First" and "second" pre- 
liminary survey. 

Location surveys 13 

15. "Paper Location." 16. Surveying methods. 17. Form of 
Notes. 

CHAPTER II. 
ALIGNMENT. 

Simple curves 18 

18. Designation of curves. 19. Length of a subchord. 20. Length of 
a curve. 21. Elements of a curve. 22. Relation between T, B, and ^. 
23. Elements of a 1° curve. 24. Exercises. 25. Curve locjition by 
deflections. 26. Instrumental work. 27. Curve location b}' two 
transits. 28. Curve location by tangential offsets. 29. Curve loca- 
tion by middle ordinates. 30. Curve location by offsets from the 
long chord. 31. Use and value of the above methods. 32. Obstacles 
to location. 33. Modifications of location. 34. Limitations in loca- 
tion. 35. Determination of the curvature of existing track. 36. 
Problems. 

V 



VI TABLE OF CONTENTS, 

PAGE 

Compound curves 37 

37. Nature aud use. 38. Mutual relations of the parts of a com- 
pound curve having two branches. 39. Modifications of location. 
40. Problems. 

Transition curves 43 

41. Superelevation of the outer rail on curves. 42. Practical rules 
for superelevation. 43. Transition from level to inclined track. 
44. Fundamental principle of transition curves. 45. Multiform com- 
pound curves. 46. Required length of spiral. 47. To find the ordi- 
nates of a l°-per-25-feet spiral. 48. To find the deflections from any 
point of the spiral. 49. Connection of spiral with circular curve aud 
with tangent. 50. Field-work. 51. To replace a simple curve by a 
curve with spirals. 52. Application of transition curves to compound 
curves. 53. To replace a compound curve by a curve with spirals. 

Vertical curves 61 

54. Necessity for their use. 55. Required length. 56. Form of 
curve. 57. Numerical example. 

CHAPTER III. 

EARTHWORK. 

Form of excavations and embankments 64 

58. Usual form of cross-section in cut and fill. 59. Terminal pyra- 
mids and wedges. 60. Slopes. 61. Compound sections. 62. Width 
of roadbed. 63. Form of subgrade. 64. Ditches. 65. Effect of 
sodding the slopes, etc. 

Earthwork surveys 72 

66. Relation of actual volume to the numerical result. 67. Pris- 
moids. 68. Cross-sectioning. 69. Position of slope-stakes. 

Computation of volume 76 

70. Prismoidal formula. 71. Averaging end areas. 72. Middle 
areas. 73. Two level ground. 74. Level sections. 75. Numerical 
example, level sections. 76. Equivalent sections. 77. Equivalent 
level sections. 78. Three-level sections. 79. Computation of prod- 
ucts. 80. Five-level sections. 81. Irregular sections. 82. Volume 
of an irregular p'ismoid. 83. True prismoidal correction for ir- 
regular prismoids. 84 Numerical example ; irregular sections ; 
volume, with true prismoidal correction. 85, Volume of irregular 
prismoid, with approximate prismoidal correction. 80. Illustration 
of value of approximate rules. 87. Cross-sectioning irregular sections. 
88. Side-hill w^ork. 89. Borrow-pits. 90. Correction for curvature. 
91. Eccentricity of the center of gravity. 92. Center of gravity of 
side-hill sections. 93. Examples of curvature correction. 94. Accu- 



TABLE OF CONTENTS. vii 

PAGE 

racy of earthwork computations. 95. Approximate compulations 
from proliles. 

Formation of embankments Ill 

9G. Shrinkage of earthwork. 97. Allowance for shrinkage. 98. 
Methods of forming embankmeuts. 

Computation of haul IIG 

99. Nature of subject. 100. Mass diagram. 101. Properties of 

the mass curve. 102. Area of the mass curve. 103. Value of the 

mass diagram. 104. Changing the grade line. 105. Limit of free 
haul. 

Cost of earthwork 136 

106. General divisions of the subject. 107. Loosening. 108. Load- 
ing. 109. Hauling. 110. Choice of method of haul dependent on 
distance. 111. Spreading. 113. Keeping roadways in order. 113. 
Repairs, wear, depreciation, and interest on cost of plant. 114. Super- 
intendeuce and incidentals. 115. Contractor's profit. 116. Limit of 
profitable haul. 

Blasting , ... 143 

117. Explosives. 118. Drilling. 119. Position and direction of 
drill-holes. 130. Amount of explosive. 121. Tamping. 133. Ex- 
ploding the charge. 123. Cost. 124. Classification of excavated 
material. 135. Specifications for earthwork. 

CHAPTER IV. 

TRESTLES. 

126. Extent of use. 137. Trestles vs. embankments. 138. Two 
principal types. 

Pile trestles , 155 

129. Pile bents. 130. Methods of driving piles. 131. Pile-driving 
formula. 133. Pile-points and pile-shoes. 133. Details of design. 
134. Cost of pile trestles. 

Framed trestles 163 

135, Typical design. 136. Joints. 137. Multiple-story construc- 
tion. 138. Span. 139. Foundations. 140. Longitudinal bracing. 
141. Lateral bracing. 143. Abutments. 

Floor systems 167 

143. Stringers. 144. Corbels. 145. Guard-rails. 14G. Ties on 
trestles. 147. Superelevation of the outer rail on curves. 148. Pro- 
tection from fire. 149. Timber. 150. Cost of framed timber 
trestles. 



Tiii TABLE OF CONTENTS, 

PAGE 

Design of wooden trestles • • 174 

151. Common practice. 152. Required elements of strength. 153. 
Strength of timber. 154, Loading. 155. Factors of safety. 156. 
Design of stringers. 157. Design of posts. 158. Design of caps and 
sills. 159. Bracing. 

CHAPTER V. 
TUNNELS. 

Surveying * • • • ^°^ 

160. Surface surveys. 161. Surveying down a shaft. 162. Under- 
ground surveys. 163. Accuracy of tunnel surveying. 

Design ^^^ 

164. Cross-sections. 165. Grade. 166. Lining. 167. Sljafts. 
168. Drains. 

Construction ^^^ 

169. Headings. 170. Enlargement. 171. Distinctive features of 
various methods of construction. 172. Ventilation duriug construc- 
tion. 173. Excavation for the portals. 174. Tunnels vs. open cuts. 
175. Cost of tunneling. \ 

CHAPTER VI. 

CULVERTS AND MINOR BRIDGES. 

176. Definition and object. 177. Elements of the design. 
Area of the waterway 20o 

178. Elements involved. 179. Methods of computation of area. 
180. Empirical formulce. 181. Value of empirical formulse. 182. 
Results based on observation. 183. Degree of accuracy required. 

Pipe culverts -^^^ 

184, Advantages. 185. Construction. 186. Iron-pipe culverts. 
187. Tile-pipe culverts. 

910 

Box culverts ^^'^ 

188. Wooden box culverts. 189. Stone box culverts. 190. Old- 
rail culverts. 

Arch culverts "15 

191. Influence of design on flow. 192. Example of arch-culvert 
design. 

Minor openings ^1^ 

193. Cattle-guards. 194. Cattle-passes. 195. Standard stringer 
and I-beam bridges. 



TABLE OF CONTENTS. IX 

CHAPTER VII. 

BALLAST. 

PAGE 

196. Purpose and requirements. 197. Materials. 198. Cross- 
sections. 199. Methods of laying ballast. 200. Cost. 

CHAPTER VIII. 

TIES 
AND OTHER FORMS OF RAIL SUPPORT. 

201. Various methods of supporting rails. 202. Economics of ties. 

007 
Wooden ties ' ^'^ 

203. Choice of wood. 204. Durability. 205. Dimensions. 206. 
Spacing. 207. Specifications. 208. Regulations for laying and 
renewing ties. 209. Cost of ties. 
Preservative processes for wooden ties 232 

210. General principle. 211. Vulcanizing. 212. Creosoting. 213. 
Burnettizing. 214. Kyanizing. 215. Wellhouse (or zinc-tannin) 
process. 216. Cost of treating. 217. Economics of treated ties. 

OQQ 

Metal ties ~^° 

218. Extent of use. 219. Durability. 220. Form and dimensions 
of metal cross-ties. 221. Fastenings. 222. Cost. 223. Bowls or 
plates. 224. Longitudinals. 

CHAPTER IX. 

RAILS. 

225. Early forms. 226. Present standard forms. 227. Weight 
for various kinds of traffic. 228. Effect of stiffness on traction. 229. 
Length of rails. 230. Expansion of rails. 231. Rules for allowing 
for temperature. 232. Chemical composition. 233. Testing. 234. 
Rail wear on tangents. 235. Rail wear on curves. 236. Cost of rails. 

CHAPTER X. 

ra il-fastenings. 

Rail-joints 255 

237. Theoretical requirements for a perfect joint. 238. Efficiency 
of the ordinary angle bar. 239. Effect of rail-gap at joints. 240. 
Supported, suspended, and bridge joints. 241. Failures of rail joints. 
242. Standard angle-bars. 243. Later designs of rail-joints. 

Tie-plates 260 

244. Advantages. 245. Elements of the design. 246. Methods of 
setting. 



X TABLE OF CONTENTS. 

PAGE 

Spikes 263 

247. Requirements. 248. Driving. 249. Screws and bolts. 250. 
Wooden spikes. 

Track-bolts and nut-locks 266 

251. Essential requirements. 252. Design of track-bolts. 253. 
Design of nut-locks. 

CHAPTER XI. 

switches and crossings. 

Switch construction 271 

254. Essential elements of a switch. 255. Frogs. 256. To find 
the frog number. 257. Stub switches, 258. Point switches. 259. 
Switch-stauds. 260. Tie-rods. 261. Guard-rails. 

Mathematical design of switches 278 

262. Design with circular lead rails. 263. Effect of straight frog- 
rails. 264. Effect of straight point-rails. 265. Combined effect of 
straight frog rails and straight point-rails. 266. Comparison of the 
above methods. 267. Dimensions for a turnout from the outer side 
of a curved track. 268. Dimensions for a turnout from the inner 
side of a curved track. 269. Double turnout from a straight track. 
270. Two turnouts on the same side. 271. Connecting curve from a 
straight track. 272. Connecting curve from a curved track to the 
outside. 273. Connecting curve from a curved track to the inside. 
274. Crossover between two parallel straight tracks. 275. Crossover 
between two parallel curved tracks. 276. Practical rules for switch- 
laying. 

Crossings 300 

277. Two straight tracks. 278. One straight and one curved track. 
279. Two curved tracks. 

Appendix. The Adjustments of Instruments 303 

Tables. 

I. Radii of curves 314 

II. Tangents and external distances to a 1° curve. 318 

III. Switch leads and distances 321 

IV. Transition curves 322 

V. Logarithms of numbers 325 

VI. Logarithmic sines and tangents of small angles 345 

Vn. Logarithmic sines, cosines, tangents, and cotangents 348 

VIII. Logarithmic versed sines and external secants 393 

IX. Natural sines, cosines, tangents and cotangents 439 

X. Natural versed sines and external secants 444 

XI. Useful trigonometrical formulae 449 

Index 451 



RAILROAD CONSTRUCTION. 



CHAPTER I. 

RAILROAD SURVEYS. 

The proper conduct of railroad surveys presupposes an 
adequate knowledge of almost the whole subject of railroad 
engineering, and particularly of some of the complicated ques- 
tions of Railroad Economics, which are not generally studied 
except at the latter part of a course in railroad engineering, if 
at all. This chapter will therefore be chiefly devoted to 
methods of instrumental work, and the problem of choosing a 
general route will be considered only as it is influenced by the 
topography or by the application of those elementary principles 
of liailroad Economics which are self-evident or which may be 
accepted by the student until he has had an opportunity of 
studying those principles in detail. 

RECONNOISSANCE SURVEYS. 

1. Character of a reconnoissance survey. A reconnoissance 
survey is a very hasty examination of a belt of country to de- 
termine which of all possible or suggested routes is the most 
promising and best worthy of a more detailed survey. It is 
essentially very rough and rapid. It aims to discover those 
salient features which instantly stamp one route as distinctly 
superior to another and so narrow the choice to routes which 
are so nearly equal in value that a more detailed survey is nec- 
essary to decide between them. 



2 BAILBOAD CONSTRVCTION. §2. 

2. Selection of a general route. The general question of 
running a railroad between two towns is usually a financial rather 
than an engineering question. Financial considerations usually 
determine that a road must pass through certain more or less 
important towns between its termini. When a railroad runs 
through a thickly settled and very flat country, where, from a 
topographical standpoint, the road may be ran by any desired 
route tlie " rio;ht-of-way agent " sometimes has a greater influ- 
ence in locating the road than the engineer. But such modifi- 
cations of alignment, on account of business considerations, are 
foreign to the engineer's side of the subject, and it will be here- 
after assumed that topography alone determines the location of 
the line. The consideration of those larger questions combin- 
ing finance and engineering (such as passing by a town on ac- 
count of the necessary introduction of heavy grades hi order to 
reach it) is likewise ignored. 

3. Valley route. This is perhaps the simplest problem. If 
the two towns to be connected lie in the same valley, it is 
frequently only necessary to run a line which shall have a nearly 
uniform grade. The reconnoissance problem consists largely in 
determining the difference of elevation of the two termini of 
this division and the approximate horizontal distance so that the 
proper grade may be chosen. If there is a large river running 
through the valley, the road will probably remain on one side 
or the other throughout the whole distance, and both banks 
should be examined by the reconnoissance party to determine 
which is preferable. If the river may be easily bridged, both 
banks may be alternately used, especially when better alignment 
is thereby secured. A river valley has usually a steeper slope 
in the upper part than in the lower part. A uniform grade 
throughout the valley will therefore require that the road climbs 
up the side slopes in the lower part of the valley. In case the 
' ' rulino- o-rade ' ' ^ for tlie whole road is as great as or greater 

* The ruling grade may liere be loosely defined as the maximum grade 
which is permissible. This definition is not strictly true, as may be seen later 
when studying Railroad Economics, but it may here serve the purpose. 



§ 5. RAILROAD SURVEYS. 3 

than the steepest natural valley slope, more freedom may be 
used in adopting' that alignment which has the least cost — 
regardless of grade. The natural slope of large rivers is almost 
invariably so low that grade has no influence in determiniTig the 
choice of location. When bridging is necessary, the river 
banks should be examined for suitable locations for abutments 
and piers. If the soil is soft and treacherous much difficultv 
may be experienced and the choice of route may be laro;elv 
determined by the difficulty of bridging the river except at 
certain favorable places. 

4. Cross-country route. A cross-country route always has one 
or more sunnnits to be crossed. The problem becomes more 
complex on account of the greater nundjer of possible solutions 
and the difficulty of jiroperly weighing the advantages and dis- 
advantages of each. The general aim should be to choose the 
lowest summits and the highest stream crossings, provided that 
by so doing the grades between these determining points shall 
be as low as possible and shall not be greater than the ruling 
grade of the road. Xearly all railroads combine cross-country 
and valley routes to some extent. Usually the steepest natural 
slopes are to be found on the cross-country routes, and also the 
greatest difficulty in securing a low through grade. An approx- 
imate determination of the ruling grade is usually made during 
the reconnoissance. If the ruling grade has been previously 
decided on by other considerations, the leading feature of the 
reconnoissance survey will be the determination of a general 
route along which it will be possible to survey a line whose 
maximum orrade shall not exceed the rulinir s-rade. 

5. Mountain route. The streams of a mountainous ree:ion 
frequently have a slope exceeding the desired ruling grade. In 
such cases there is no possibility of securing the desired grade 
by following the streams. The penetration of such a region 
may only be accomplished by "development" — accompanied 
perhaps by tunneling. "Development" consists in deliber- 
ately increasing the length of the road between two extremes 
of elevation so that the rate of grade shall be as low as desired. 



SAILBOAD CONSTBTICTION. 



§5. 



The usual method of accomphshing this is to take advantage of 
some convenient formation of the ground to introduce some 
lateral deviation. The methods may be somewhat classified as 
follows : 

(a) Running the line up a convenient lateral valley, turning 
a sharp curve and working back up the opposite slope. As 
shown in Fig. 1, the considerable rise between ^ and ^ was 




Fig. 1. 

surmounted by starting off in a very different direction from 
the general direction of the road ; then, when about one-half of 
the desired rise had been obtained, the line crossed the valley 
and continued the climb along the opposite slope, (b) Switch- 
hack. On the steep side-hill BCD (Fig. 1) a very considerable 
gain in elevation was accomplished by the switchback CD. 
The gain in elevation from B \o D \& very great. On the 
other hand, the speed must always be slow ; there are two com- 
plete stoppages of the train for each run ; all trains must run 
backward from C to D. (c) Bridge spiral. When a valley is 
so narrow at some point that a bridge or viaduct of reasonable 
length can span the valley at a considerable elevation above the 
bottom of the valley, a bridge spiral may be desirable. In 



6. 



RAILROAD SURVEYS. 



b 



Fig. 2 tlie line ascends tlic stream valley past yl, crosses tlie 
stream at B, works back to the narrow ])lace at C\ and there 
crosses itself, having gained perhaps 100 feet in elevation, 
(d) Tunnel spiral. This is the reverse of the previous plan. 





V, 



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.\^^ 



Fig. 3. 



Fig. 3 



It implies a thin steep ridge, so thin at some place that a tunnel 
through it will not be excessively long. Switchbacks and 
spirals are sometimes necessary in mountainous countries, but 
they should not be considered as normal types of construction. 
A region must be very difficult if these devices cannot be 
avoided. 

Rack railways and cable roads, although types of mountain 
railroad construction, will not be here considered. 

6. Existing maps. The maps of the U. S. Geological Survey 
are exceedingly valuable as far as they have been completed. 
So far as topographical considerations are concerned, they 
almost dispense with the necessity for the reconnoissance and 
*' first preliminary " surveys. Some of the State Survey maps 
will give practically the same information. County and town- 
ship maps can often be used for considerable information as to the 
relative Jiorizoiital position of governing points, and even some 



6 RAILROAD CONSTRVCTION, § 7. 

approximate data regarding eleA^ations may be obtained bj a 
study of the streams. Of course such information will not dis- 
pense with surveys, but will assist in so planning them as to 
obtain the best information with the least work. AVhen the 
relative horizontal positions of points are reliably indicated on a 
map, the reconnoissance may be reduced to the determination 
of the relative elevations of the governing points of the route. 

7. Determination of relative elevations. A recent description 
of European methods includes spirit-leveling in the reconnois- 
sance work. This may be due to the fact that, as indicated 
above, previous topographical surveys have rendered unnecessary 
the " exploratory " survey which is required in a new country, 
and that their reconnoissance really corresponds more nearly to 
our preliminary. 

The perfection to which barometrical methods have been 
brought has rendered it possible to determine differences of 
elevation with sufficient accuracy for reconnoissance purposes 
by the combined use of a mercurial and an aneroid barometer. 
The mercurial barometer should be kept at " headquarters," and 
readings should be taken on it at such frequent intervals that 
any fluctuation is noted, and throughout the period that observa- 
tions with the aneroid are taken in the field. At each observa. 
tion there should also be recorded the time, the readinsr of the 
attached thermometer, and the temperature of the external 
air. For uniformity, the mercurial readings should then be 
''reduced to 32° F." Before starting out, a reading of the 
aneroid should be taken at headquarters coincident with a read- 
ing of the mercurial. The difference is one value of the correc- 
tion to the aneroid. As soon as the aneroid is brouo-ht back 
another comparison of readings should be made. Even though 
there has been considerable rise or fall of pressure in the interval, 
the difference in readings (the correction) should be substantially 
the same provided the aneroid is a good instrument. The best 
aneroids read directly to ^^ of an inch of mercury and may be 
estimated to yo^qt ^^ ^^ iiioh — which corresponds to about 0.9 
foot dift'erence of elevation. In the field there should be read, 



§ 8. RAILROAD SURVEYS. 7 

at each point whose elevation is desired, tlie aneroid, the time, 
-and tlie temperature. These readings, corrected by the mean 
vahie of the correction between the aneroid and the mercurial, 
should then be combined with tlie reading of the mercurial 
(interpolated if necessary) for the times of the aneroid observa- 
tions and the difference of elevation obtained. [See the autlior's 
*' Problems in the Use and Adjustment of Engineering In- 
struments," Prob. 22.] Important points should be observed 
more than once if possible. Such duplicate observations will be 
found to give surprisingly concordant results even when a 
general fluctuation of atmospheric pressure so modifies the 
tabulated readings that an agreement is not at first apparent. 
Yariations of pressure produced by high Avinds, thunder-storms, 
■etc., will generally vitiate possible accuracy by this metliod, 
J3y ' ' headquarters ' ' is meant any place wliose elevation above 
a,ny given datum is known and where the mercurial may be 
placed and observed while observations within a range of several 
miles are made with the aneroid. If necessary the elevation of 
a new headquarters may be determined by the above method, 
but there sfiould be if possible several independent observations 
whose accordance will give a fair idea of tlieir accuracy. 

The above method should be neither sliglited nor used for 
more than it is worth. Wlien properly used, the errors are 
compensating rather than cumulative. Wlien used, for example, 
to determine that a pass J^ is 260 feet higlier than a determined 
bridge crossing at A which is six miles distant, and tliat another 
pass 6^ is 310 feet higher than A and is ten miles distant, the 
figures, even with all necessary allowances for inaccuracy, will 
give an engineer a good idea as to the choice of route especially 
as affected by ruling grade. There is no comparison between 
the time, and labor involved in obtaining the above information 
by barometric and by spirit-leveling methods, and ybr recon- 
noissance purposes the added accuracy of the spirit-leveling 
method is hardly worth its cost. 

8. Horizontal measurements, bearings, etc. AVhen there is 
no map which may be depended on, or when only a skeleton 



8 RAILROAD CONSTRUCTION. % 9. 

map is obtainable, a rapid survey, sufficiently accurate for the 
purpose, may be made by using a pocket compass for bearings 
and a telemeter, odometer, or pedometer for distances. The 
telemeter [stadia] is more accurate, but it requires a definite clear 
sight from station to station, which may be difficult through a 
wooded country. The odometer, which records the revolutions 
of a wheel of known circumference, may be used even in rough 
and wooded country, and the results may be depended on to a 
small percentage. The pedometer (or ^^ace -measurer) depends 
for its accuracy on the actual movement of the mechanism for 
each pace and on the uniformity of the pacing. Its results are 
necessarily rough and approximate, but it may be used to fill 
in some intermediate points in a large skeleton map. A hand- 
level is also useful in determining the relative elevation of various 
topographical features which may have some bearing on the 
proper location of the road. 

9. Importance of a good reconnoissance. The foregoing in- 
struments and methods should be considei-ed only as aids in 
exercising an educated common sense, without which a proper 
location cannot be made. The reconnoissance survey should com- 
mand the best talent and the greatest experience available. 
If the general route is properly chosen, a comparatively low 
order of engineering skill can fill in a location which will prove 
a paying railroad property ; but if the general route is so chosen 
that the ruling grades are high and the business obtained is small 
and subject to competition, no amount of perfection in detailed 
alignment or roadbed construction can make the road a profitable 
investment. 

PRELIMINAKT SURVEYS. 

10. Character of survey. A preliminary railroad survey is 
properly a topographical survey of a belt of country which has 
been selected during the reconnoissance and within which it is 
estimated that the located line will lie. The width of this belt 
will depend on the character of the country. When a railroad 



§11 



RAILROAD SURVEYS. 



9 



is to follow a river having very steep banks the choice of loca- 
tion is sometimes limited at places to a very few feet of width 
and the belt to be surveyed may I)e correspondingly narrowed. 
In very flat coimtry the desired width may be only limited by the 
ability to survey points with sufficient accuracy at a considera])le 
distance from w^hat may be called the "backbone line" of the 
survey. 

11. Cross-section method. This is the only feasible method 
in a wooded country, and is employed by many for all kinds 
of country. The hackbone line is surveyed either by observ- 
ing magnetic bearings with a compass or by carrying forward 




Fig. 4. 



absolute azimuths with a transit. The compass method lias 
the disadvantages of hmited accuracy and the possibility of 



10 RAILROAD CONSTRUCTION. § 12. 

considerable local error owing to local attraction. On the other 
hand there are the advantages of greater simplicity, no necessity 
for a back rodman, and tlie fact that the errors are purely 
local and not cumulative, and may be so limited, with care, that 
they will cause no vital error in the subsequent location survey. 
The transit method is essentially more accurate, but is liable 
to be more laborious and troublesome. If a large tree is 
encountered, either it must be cut down or a troublesome opera- 
tion of offsetting must be used. If the compass is employed 
under these circumstances, it need only be set up on the far side 
of the tree and the former bearing produced. An error in 
reading a transit azimuth will be carried on throughout the 
survey. An error of only five minutes of arc will cause an off- 
set of nearly eight feet in a mile. Large azimuth errors may, 
however, be avoided by immediately checking each new azimuth 
with a needle reading. It is advisable to obtain true azimuth 
at the beginning of the survey by an observation on the sun or 
Polaris, and to check the azimuths every few miles by azimuth 
observations. Distances along the backbone line should be 
measured with a chain or steel tape and stakes set every 100 
feet. When a course ends at a substation, as is usually the case, 
the remaining portion of the 100 feet should be measured along 
the next course. The level party should immediately obtain the 
•elevations (to the nearest tenth of a foot) of all stations, and also 
<of the lowest points of all streams crossed and even of dry gullies 
^hicli would require culverts. 

12. Cross-sectioning. It is usually desirable to obtain con- 
tours at five-foot intervals. This may readily be done by the 
use of a Locke level (w'hich should be held on top of a simple 
five-foot stick), a tape, and a rod ten feet in length graduated 
to feet and tenths. The method of use may perhaps be best 
explained by an example. Let Fig. 5 represent a section per- 
pendicular to the survey line — such a section as would be made 
by the dotted lines in Fig. 4. C represents the station point. 
Its elevation as determined by the level is, say, 158.3 above 
datum. When the Locke level on its five-foot rod is placed at 



§12. 



RAILROAD SURVEYS. 



11 



C^ the level has an elevation of 163.3. Therefore when a point 
is found (as at a) where the level will read 3.3 on the rod, that 




Fig. 5. 



point has an elevation of 160.0 and its distance from the center 
gives the position of the 160-foot contour. Leaving the long 
rod at that point (rt), carry the level to some point (b) such that 
the level will sight at the toj> of the rod. h is then on the 165- 




FiG. 6. 

foot contour, and the horizontal distance ah added to the liori- 
zontal distance ac gives the position of that contour from tlie 
center. The contours on the lower side are found similarly. 
The first rod reading will he 8.3, giving the 155-foot contour. 
Plot the results in a note-book which is ruled in cpiarter-inch 
squares, using a scale of 100 feet per inch in both directions. 



12 BAILROAD CONSTRUCTION. § 13. 

Plot tlie work up the page ; then when looking ahead along the 
line, the work is properly oriented. When a contonr crosses 
the survey line, the place of crossing may be similarly deter- 
mined. If the ground flattens out so tliat five-foot contours are 
very far apart, the absolute elevations of points at even fifty- 
foot distances from the center should be determined. The 
method is exceedingly rapid. Whatever error or inaccuracy 
occurs is confined in its effect to the oue station where it occurs. 
The work being thus plotted in the field, unusually irregular 
topography may be plotted with greater certainty and no great 
error can occur without detection. It would even be possible 
by this method to detect a gross error that might have been 
made by the level party. 

13. Stadia method. This method is best adapted to fairly 
open country where a "shot" to any desired point may be 
taken without clearing. The haclcbone survey line is the same 
as in the previous method except that each course is limited to 
the practicable length of a stadia sight. The distance between 
stations should be checked by foresight and backsight — also the 
vertical angle. Azimuths should be checked by the needle. 
Considering the vital importance of leveling on a railroad survey 
it might be considered desirable to run a line of levels over the 
stadia stations in order that the leveling may be as precise as 
possible ; but when it is considered that a preliminary survey is 
a somewhat hasty survey of a route that onay be abandoned, and 
that the errors of leveling by the stadia method (which are com- 
pensating) may be so minimized that no proposed route would 
be abandoned on account of such small error, and that the effect 
of such an error may be easily neutralized by a slight change in 
the location, it may be seen that excessive care in the leveling 
of the preliminary survey is hardly justifiable. 

Since the students taking this work are assumed to be familiar 
with the methods of stadia topographical surveys, this j^art of 
the subject will not be further elal)orated. 

14. " First " and " second " preliminary surveys. Some engi- 
neers advocate two preliminary surveys. When this is done, 



§15. RAILROAD SURVEYS. 13 

the first is a very rapid survey, made perhaps witli a compass, 
and is only a better grade of reconnoissance. Its aim is to 
rapidly develop the facts which will decide for or against any 
proposed route, so that if a route is found to be unfavorable 
another more or less modified route may be adopted without 
having wasted considerable time in the survey of useless details. 
By this time the student should have grasped the fundamental 
idea that both the reconnoissance and preliminary surveys are 
not surveys of li)ies but of areas \ that their aim is to survey 
only those topographical features which would have a deter- 
mininof influence on anv railroad line which mii^ht be constructed 
through that particular territory, and that the vahie of a locating 
engineer is largely measured by his ability to recognize those 
'determining influences with the least amount of work from his 
-surveying corps. Frequently too little time is spent on the 
comparative study of ]u'eliminary lines. A line will be hastily 
decided on after very little study ; it will then be surveyed with 
minute detail and estimates carefully worked up, and the claims 
of any other suggested route will then be handicapped, if not 
disregarded, owing to an unwillingness to discredit and throw 
away a large amount of detailed surveying. The cost of two or 
three extra preliminary surveys {at critical 2>oints and not over 
the whole line) is utterly insignificant compared with the j^rob- 
able improvement in the "operating value" of a line located 
after such a comparative study of preliminary lines. 

LOCATION SURVEYS. 

15. "Paper location." When the preliminary survey has 
been plotted to a scale of 200 feet per inch and the contours 
drawn in, a study may be made for the location survey. Disre- 
garding for the present the effect on location of transition curves, 
the alignment may be said to consist of straight lines (or " tan- 
gents ") and circular curves. The " paper location '* therefore 
consists in plotting on the preliminary map a succession of 
straight lines which are tangent to the circular curves connect- 



14 RAILROAD CONSTRUCTION. § 15. 

ing tliem. Tlie determining points should first be considered. 
Such points are the termini of the road, the lowest practicable 
point over a summit, a river-crossing, etc. So far as is possi- 
ble, having due regard to other considerations, the road should 
be a ''surface" road, i.e., the cut and fill should be made as 
small as possible. The maximum permissible grade must also 
have been determined and duly considered. The method of 
location differs radically according as the lines joining the deter- 
mining points have a very low grade or have a grade that ap- 
proaches the maximum permissible. With very low natural 
grades it is only necessary to strike a proper balance between 
the requirements for easy alignment and the avoidance of exces- 
sive earthwork. When the grade between two determined 
points approaches the maximum, a study of the location may be 
begun by finding a strictly surface line which will connect those 
points with a line at the given grade. For example, suppose 
the required grade is 1.6^ and that the contours are drawn at 
5-foot intervals. It will require 312 feet of 1.6^ grade to rise 
5 feet. Set a pair of dividers at 312 feet and step off this in- 
terval on successive contours. This line will in general be very 
irregular, but in an easy country it may lie fairly close to the 
proper location line, and even in difficult country such a surface 
line will assist greatly in selecting a suitable location. When the 
larger part of the Kne will evidently consist of tangents, the tan- 
gents should be first located and should then be connected by 
suitable curves. When the curves predominate, as they gener- 
ally will in mountainous country, and particularly when the line 
is purposely lengthened in order to reduce the grade, the curves 
should be plotted first and the tangents may then be drawn 
connecting them. Considering the ease with which such lines 
may be drawn on the preliminary map, it is frequently advisable, 
after making such a paper location, to begin all over, draw a 
new line over some specially difficult section and compare re- 
sults. Profiles of such lines may be readily drawn by noting their 
intersection with each contour crossed. Drawing on each profile 
the required grade line will furnish an approximate idea of the 



§ 16. BAILROAD SURVEYS, 15 

coTTiparative amount of earthwork required. After deciding on 
the paper location, the length of each tangent, the central angle 
(see § 21), and the radius of each curve sliould be measured as 
accurately as possible. Since a slight error made in such meas- 
urements, taken from a map with a scale of 200 feet per inch, 
would by accumulation cause serious discrepancies between the 
plotted location and the location as afterward surveyed in the 
field, frequent tie lines and angles should be determined between 
the ])lotted location line and the preliminary line, and the loca- 
tion should be altered, as may prove necessary, by changing the 
length of a tangent or changing the central angle or radius of a 
curve, so that the agreement of the check-points will be suffi- 
ciently close. The errors of an inaccurate preliminary survey 
may thus be easily neutralized (see § 33). When the pre- 
liminary line has been properly run, its "backbone" line will 
lie very near the location line and will probably cross it at fre- 
quent intervals, thus rendering it easy to obtain short and nu- 
merous tie lines. 

16. Surveying methods. A transit should be used for align- 
ment, and only precise work is allowable. The transit stations 
should be centered with tacks and should be tied to witness- 
stakes, which should be located outside of the range of the earth- 
work, so that they will neither be dug up nor covered up. All 
original property lines lying within the limits of the right of way 
should be surveyed with reference to the location line, so that 
the right-of-way agent may have a proper basis for settlement. 
"When the property lines do not extend far outside of the re- 
quired right of way they are frequently surveyed completely. 

The leveler usually reads the target to the nearest thousandth 
of a foot on turning-points and bench-marks, but reads to the 
nearest tenth of a foot for the elevation of the ground at 
stations. Considering that yif-oir ^^ ^ ^^^^ ^^^^ ^^^ angular value 
of only 7 seconds at a distance of 300 feet, and that one division 
of a level-bubble is usually about 30 seconds, it may be seen that 
it is a useless refinement to read to thousandths unless corre- 
sponding care is taken in the use of the level. The leveler 



16 RAILROAD CONSTRUCTION. § 17. 

should also locate liis bench-marks outside of the range of 
earthwork. A knob of rock protruding from the ground affords 
an excellent mark. A large nail, driven in the roots of a tree, 
which is not to be disturbed, is also a good mark. These marks 
should be clearly described in the note-book. The leveler should 
obtain the elevation of the ground at all station-points ; also at 
all sudden breaks in the profile line, determining also the distance 
of these breaks from the previous even station. This will in- 
clude the position and elevation of all streams, and even dry 
gullies, which are crossed. 

Measurements should preferably be made with a steel tape, 
care being taken on steep ground to insure horizontal measure- 
ments. Stakes are set each 100 feet, and also at the beginning 
and end of all curves. Transit-points (sometimes called " plugs " 
or "hubs") should be driven flush with the ground, and a 
" witness- stake," having tlie "number" of the station, should 
be set three feet to the right. For example, tlie witness-stake 
might have on one side " 137 + 69.92," and on the other side 
" P C 4° K," which would signify that the transit hub is 69.92 
feet beyond station 137, or 13769.92 feet from the beginning of 
the line, and also that it is the "point of curve" of a " 4°- 
curve ' ' which turns to the right. 

Alignment. The alignment is evidently a part of the loca- 
tion survey, but, on account of the magnitude and importance 
of the subject, it will be treated in a separate chapter. 

17. Form of Notes. Although the Form of ]N"otes cannot be 
thoroughly understood until after curves are studied, it is nere 
introduced as being the most convenient place. The right-hand 
page should have a sketch showing all roads, streams, and 
property lines crossed with the bearings of those lines. This 
should be drawn to a scale of 100 feet per inch — the quarter- 
inch squares which are usually ruled in note-books giving con- 
venient 2 5 -foot spaces. This sketch will always be more or less 
distorted on curves, since the center line is always shown as 
straight regardless of curves. The station points ("Sta." in 
first column, left-hand page) should be placed opposite to their 



§17. 



RAILROAD SURVKYS. 



17 



sketched positions, which means that even stations will be 
recorded on every fourth line. This allows three intermediate 
lines for substations, which is ordinarily more than sufficient. 
The notes should read up the page, so that the sketch will be 
properly oriented when looking ahead along the line. The 
other columns on the left-hand page will be self-explanatory 
when the subject of curves is understood. If the ' ' calculated 
bearings ' ' are based on azimuthal observations, their agreement 
(or constant diiference) with the needle readings will form a 
valuable check oh the curve calculations and the instrumental 
work. 

FORM OF NOTES. 
[Left-hand papre.] [Right-hand page.] 



Sta. 



54 

53 
0+72.2 

52 

51 

O 50 

49 

48 

0-1-32 
47 

46 



Align- 
ment 



Vernier 



P.T. 






P.O. 



9° 11' 
7 57 

6 15 

4 33 

2 51 

1 09 
0° 



Tang. 
Defl. 



18° 22' 



Calc. 
Bearing. 



N 54° 48' E 



N 36° 26' E 



NeedU 



N 6S° 15' 1 



N 14° 0' ]■ 




CHAPTEK II. 



ALIGNMENT. 



In this chapter the alignment of the center line only of a 
pair of rails is considered. When a railroad is crossing a sum- 
mit in the grade line, altliough the horizontal projection of the 
alignment may be straight, the vertical projection will consist of 
two sloping lines joined by a cnrve. When a curve is on a 
grade, the center line is really a spiral, a curve of double curva- 
ture, although its horizontal projection is a circle. The center 
line therefore consists of straight lines and curves of single 
and double curvature. The simplest method of treating them 
is to consider their horizontal and vertical projections separately. 
In treating simple, compound, and transition curves, only the 
horizontal projections of those curves will be considered. 



SIMPLE CURVES. 



18. Designation of curves. A curve may be designated 

either by its radius or by the angle 
subtended by a chord of unit length. 
Such an angle is known as the ' ' degree 
of curve ' ' and is indicated by D. 
Since the curves that are practically 
used have very long radii, it is gener- 
ally impracticable to make any use of 
the actual center, and the curve is 
located without reference to it. If 
AB in Fig. 7 represents a unit chord 
((7) of a curve of radius i?, then by the above defini- 

18 




§19. 



ALIGNMENT. 



19 



tion the angle AOB equals D, Then AO sin ^D = iAB ^ 

iO. 

(1) 



.-. B = 



or, bj inversion, 



sin ^J) = 



sin il) 

C_ 
2E 



(2) 



The unit chord is variously taken throughout the world as 
100 feet, (S^ feet, and 20 meters. In the United States 100 
feet is invariably used as the unit chord length, and throughout 
this work it will be so considered. Table I has been computed 
on this basis. It gives the radius, with its logarithm, of all 
curves from a 0° 01' curve up to a 10° curve, varying by single 
minutes. The sharper curves, which are seldom used, are given 
with larger intervals. 

An approximate value of i? may be readily found from the 
following simple rule, which should be memorized : 



B = 



5730 

IT' 



Although such values are not mathematically correct, since jR 
does not strictly vary inversely as D, yet the resulting value is 
within a tentli of one per cent for all 
commonly used values of ^, and is suf- 
ficiently close for many purposes, as will 
be shown later. 

19. Length of a sub-chord. Since 
it is impracticable to measure along a 
curved arc, curves are always measured 
by laying off 100-foot chord lengths. 
Tliis means that the actual arc is always 
a little longer than the chord. It also 
means that a suhchord (a chord shorter than the unit length) 
will be a little longer than the ratio of the angles subtended 
would call for. The truth of this may be seen without calcu- 




FiG. 8. 



20 RAILROAD CONSTRUCTION. § 20. 

lation by noting that two equal subcliords, each subtending the 
angle j-T), will evidently be slightly longer than 50 feet each. 
If c be the length of a subchord subtending the angle d, then, 
as in Eq. (2), 



sm 2,ct — Q~o) 



or, by inversion, 



c=^2B sin ^d (3) 



d 
The no?ninal length of a subchord = 100—. For example,, 

a nominal subchord of 40 feet will subtend an angle of -^-^q of 
D° ; its true length will be slightly more than 40 feet, and may 
be computed by Eq. 3. The difference between the nominal 
and true lengths is maximum when the subchord is about 57 
feet long, but with the low degrees of curvature ordinarily used 
the difference may be neglected. "With a 10° curve and a 
nominal chord length of 60 feet, the true length is 60.049 feet. 
Very sharp curves should be laid off with 50-foot or even 25- 
foot chords (nominal length). In such cases especially the true 
lengths of these subcliords should be computed and used instead 
of the nominal lengths. 

20. Length of a curve. The length of a curve is always 
indicated by the quotient of 100/^ -^ D. If the quotient of 
z/ -=- Z^ is a whole number, the length as thus indicated is the 
true length — measured in 100-foot choi'd lengths. If it is an 
odd number or if the curve begins and ends with a subchord 
(even though A -^ D \& a whole number), theoretical accuracy 
requires that the true subchord lengths shall be used, although 
the difference may prove insignificant. The length of the arc 
(or the mean length of the two rails) is therefore always in 
excess of the length as given above. Ordinarily the amount 
of this excess is of no practical importance. It simply adds an 
insignificant amount to the length of rail required. 

Examjple. Required the nominal and true lengths of a 
3° 45' curve having a central angle of 17° 25'. First reduce 



§22. 



ALIGNMENT. 



21 



the degrees and minutes to decimals of a degree. (100 X 1T° 25') 
-h 3° ^5' = 17^:1.007 -=- 3.75 = U^AU. The curve has four 
100-foot chords and a nominal chord of Gir.-l-ll:. The true 
chord should be 61:. 451. The actual arc is 



17M:1G7 X 



7t 



IbO' 



X E = 461:. 527. 



The excess is therefore 46-1.527 - 464.451 = 0.076 foot. 

21. Elements of a curve. Considering the line as running 
from A toward I>^ tlie beginning of tlie curve, at A^ is called 
i\\Q point of curve {PC). The other end of the curve^ at ^, is' 
called the point of tangency (PT). 
The intersection of the tangents is 
called the vertex {V). The angle 
made bj the tangents at T", which 
equals the angle made by the radii to 
the extremities of the curve, is called 
the central angle [A). A T^and B T", 
the two equal tangents from tlie vertex 
to the PC and PT^ are called the 
tangent distances {T). The chord 
AB is called the long chord (LC). 
The intercept HG from the middle 
of the long chord to the middle of the arc is called the middle 
ordinate (M). That part of the secant G V from the middle of 
the arc to the vertex is called the external distance (E), From 
the figure it is very easy to derive the follow^ing frequently used 
relations : 




Fig. 9. 



T= R tan ^ J 
LC 
M 



2^ sin i J 



E = 



(4) 

(5) 

R vers ^z/ (6) 

R exsec ^A (7) 



22. Relation between T, E, and A. Join A and G in Fig. 9. 
The angle VAG = iA, since it is measured by one half of the 



22 RAILROAD CONSTRUCTION. § 23. 

arc AG between the secant and tangent. AGO z= 90° —\A. 
AY: VG:\miAGV:dn YAG) 
miAGY=^ QinAGO — cosiz?; 

T'.Ew cos \A : sin \A ; 
T=Eq,oI\A (8) 



The same relation may be obtained by dividing Eq. 4 by Eq. 7, 
since tan a -^ exsec a — cot \a. 

23. Elements of a 1^ curve. From Eqs. 1 to 8 it is seen that 
the elements of a curve vary directly as R. It is also seen to 
be very nearly true that B, varies inversely as D. If the ele- 
ments of a 1° curve for various central angles are calculated and 
tabulated, the elements of a curve of Z^° curvature may be 
approximately found by dividing by D the corresponding elements 
of a 1° curve having the same central angle. For small central 
angles and low degrees of curvature the errors involved by the 
approximation are insignificant, and even for larger angles the 
errors are so small \h2Xf0r many jpurposes they may be disre- 
garded. 

In Table II is given the value of the tangent distances, 
external distances, and long chords for a I'' curve for various 
central angles. The student should familiarize himself with the 
degree of approximation involved by solving a large number of 
cases under various conditions by the exact and approximate 
methods, in order that he may know when the approximate 
method is sufficiently exact for the intended purpose. The 
approximate method also gives a ready check on the exact 
method. 

24. Exercises, [a) "What is the tangent distance of a 4° 20' 
curve having a central angle of 18° 24' ? 

{h) Given a 3° 30' curve and a central angle of 16° 20', 
how far will the curve pass from the vertex ? [Use Eq. 7.] 

(c) An 18° curve is to be laid off using 25-foot (nominal) 
chord lengths. What is the true length of the subchords ? 



§25. 



ALIGNMENT. 



23 



{(l) Given two tangents making a central angle of 15° 2^'. 
It is desired to connect these tangents by a curve which shall 
j^ass 16.2 feet from their intersection. How far down the 
tangent will the curve begin and what will be its radius ? (Use 
Eq. S and then use Eq. -i inverted.) 

25. Curve location by deflections. The ano-le between a 
secant and a tangent (or between two secants intersecting on an 
arc) is measured by one half of the intercepted arc. Beginning 
at the PC {A in Fig. 10), if the first chord is to be a full cliord 
we may deflect an angle VAa [= jP), 
and the point «, which is 100 feet from 
^•1, is a point on the curve. For the 
next station, ^, deflect an additional 
angle hAa {— ^D) and, with one end 
of the tape at a, swing the other end 
until the 100-foot point is on the line 
Ah. The points is then on the curve. 
If the final chord cjB is a subchord, its 
additional deflection {^a) is something 
less than 4-7>. The last deflection 
{BA Y) is of course ^//. It is particularly inqwrtant, when a 
curve begins or ends with a subchord and the defiections are 
odd quantities, that the last additional defiection should be care- 
fully coni})uted and added to the previous deflection, to check 
the mathematical work by the agreement of this last conqnited 
deflection with -g-z/. 

Example. Given a 3° 2-1' curve having a central angle of 
18° 22' and beginning at sta. -IT -\- 32, to conq^ute the deflections. 
The nominal length of curve is 18° 22'- 3° 24' =18.367 — 
3.40 = 5.402 stations or 540.2 feet. The curve therefore ends 
at sta. 52 + 72.2. The deflection for sta. 48 is y^o X K^^ ^^0 
= 0.68 X 1°.T = 1°.156 = r 09' nearly. For each additional 
100 feet it is 1° 42' additional. The final additional deflection 
for the final subchord of 72.2 feet is 




Fig. 10. 



^^ X K3° 24') = 1°. 2274 
100 ^^ ^ 



1° 14' nearly. 



24 RAILROAD CONSTRUCTION. §26. 

The defections are 

P. C . . . . Sta. 47 + 32 0° 

48 0° + 1° 09' = 1° 09' 

49 1° 09'+ 1° 42' = 2" 51' 

50 2° 51'+ 1° 42' r=4° 33' 

51 4° 33'+ 1° 42' ==G° 15' 

52 0° 15'+ 1° 42' = 7° 57' 

P. T 52 + 72.2 . . . /7° 57' + 1° 14^ = 9° 11' 

As a check 9° 11' = i(18° 22') = |^. (See the Form of Notes 
in § 17.) 

26. Instrumental work. It is generally impracticable to 
locate more than 500 to 600 feet of a curve from one station. 
Obstructions will sometimes require that the transit be moved up 
every 200 or 300 feet. There are two methods of setting off 
the angles when the transit has been moved up from the PC. 

(a) The transit may be sighted at the previous transit station 
with a reading on the plates equal to the deflection angle from 
that station to the station occupied, but with the angle set oif on 
the other side of 0°, so that when the telescope is turned to 0° it 
will sight along the tangent at the station occupied. Plunging 
the telescope, the forward stations may be set off by deflecting 
the proper deflections from the tangent at the station occupied. 
This is a very common method and, when the degree of curva- 
ture is an even number of degrees and when the transit is only 
set at even stations, there is but little objection to it. But the 
degree of curvature is sometimes an odd quantity, and the exi- 
gencies of difiicult location frequently require that substations 
be occupied as transit stations. Method {a) will then require 
the recalculation of all deflections for each new station occupied. 
The mathematical work is largely increased and the probability 
of error is very greatly increased and not so easily detected. 
Method {!)) is just as simple as method (a) even for the most 
simple cases, and for the more difiicult cases just referred to the 
superiority is very great. 



§26. 



ALIGNMENT. 



26 



(b) Calculate the deflection for each station and substation 
throughout the curve as though the whole curve were to be lo- 
cated from the PC. The computations may thus be completed 
and checked (as above) before beginning the instrumental work. 
If it unexpectedly becomes necessary to introduce a substation 
at any point, its deflection from the P(7may be readily inter- 
polated. The stations actually set from the PC are located as 
usual. Rule. When the transit is set on any forward station, 
backsight to ANY previous station with the plates set at the deflec- 
tion angle for the station sighted at. Plunge the telescope and 
sight at any forward station with the deflection angle originally 
computed for that station. AVlien the plates read the deflection 
angle for the station occupied, the telescope is sighting along the 
tangent at that station — which is the method of getting the for- 
ward tangent when occupying the PT. Even though the sta- 
tion occupied is an unexpected substation, 
when the instrument is properly oriented at 
that station, the angle reading for any station, ^^ 
forward or back, is that originally computed 
for it from the P(7. In diflicult work, where 
there are obstructions, a valuable check on 
the accuracy may be found by sighting back- 
ward at any visible station and noting whether 
its deflection agrees with that originally com- 
puted. As a numerical illustration, assume 
a -t° curve, with 28° curvature, with stations 
0, 2, 4, and T occupied. After setting 
stations 1 and 2, set up the transit at sta. 
2 and backsight to sta. with the deflection 
for sta. 0, which is 0°. The reading on sta. 
1 is 2° ; when the reading is 4° the telescope 
is tangent to the curve, and when sighting 
at 3 and 4 the deflections will be 6° and 8°. 
Occupy 4 ; sight to 2 with a reading of 4°. 
is 8° the telescope is tangent to the curve and, by plunging the 
telescope, 5, 6, and 7 may be located with the originally com- 




FlG. 11. 



When the reading 



26 



RAILROAD CONSTRUCTION. 



27 



puted deflections of 10°, 12°, and 14°. When occupying 7 ?, 
backsight may be taken to any visible station with the plates read 
ing the deflection for that station ; then when the plates read 
14° the telescope will point along the forward tangent. 

The location of curves by deflection angles is the normal 
method. A few other methods, to be described, should be con- 
sidered as exceptional. 

27. Curve location by two transits. A curve might be located 
more or less on a swamp where accurate chaining would be ex- 
ceedingly difficult if not impossible. The long chord AB may 
be determined by triangulation or otherwise, and the elements of 





Fig. 13. 



Fig. 13. 



the curve computed, including (possibly) subchords at each end. 
The deflection from A and B to each point may be computed. 
A rodman may then be sent (by whatever means) to locate long 
stakes at points determined by the simultaneous sightings of the 
two transits. 

28. Curve location by tangential offsets. When a curve is 
very flat and no transit is at hand the following method may be 



§ 29. ALIGNMENT. 27 

used : Produoe the back tangent as far forward as necessary. 
Compute the ordinates Oa\ Oh\ Oc\ etc., and the abscissae a' a., 
h'h, c'c. etc. If Oa is a full station (100 feet), then 

Oa' = Oa' =^ 100 cosiD, also = R' sin D; ] 

01/ = Oa'Ara'h' = 100 cos *j9+ 100 cos |/>, 

also — li sin 'ID ; L p^ 
Oc' = Oa +a'h' -\-h'c' =^100{cos>il) +C06 U) + co^^I))j 

also — Ic sin 3D ; 
etc. 

a' a = 100 sin ^Z>, also = /*' vers 7>; ^ 

h"b = a:' a + h''h =100 sin ii> + 100 sin :] />, | 

also = J^vers2D; I mq) 
c'c' = Jj'h + c'c = 100(siniZ) + sin|Z> + sinfZ>), 

also — I^versoD] 
etc. 

The functions Ji>, fi^, etc., may be more conveniently used 
without logarithms, by adding the several national trigonometrical 
functions and pointing off two decimal places. It may also be 
noted that oV (for example) is one half of the long chord 
for four stations; also that h'h is the middle ordinate for four 
stations. If the engineer is provided with tables giving the long 
chords and middle ordinates for various degrees of curvature, 
these quantities may be taken (perhaps by interpolation) from 
such tables. 

If the curve begins or ends at a substation, the angles and 
terms will be correspondingly altered. The modifications may 
be readily deduced on the same principles as above, and should 
be worked out as an exercise by the student. 

29. Curve location by middle ordinates. Take first the simpler 
case when the curve begins at an even station. If we consider 
(in Fig. 14) the curve produced back to ^, the chord za = 
2 X 100 cos iD, A'a = 100 cos iZ>, and A' A = am = .iit = 
100 sin ^D. Set off A A' perpendicular to the tangent and 
A'a parallel to the tangent. ^1^1' = aa' = hh' = cc\ etc. — 
100 sin \D. Set ofi aa' per^^endicular to a' A. Produce Aa' 



28 



BAILROAD COISSTRUCTION. 



§30. 



until a'h =-■ A'a^ thus determining h. Succeeding points of the 
curve may thus be determined indefinitely. 

Suppose the curve begins with a subchord. As before 
ra = Am' = c' cos \d' ^ and rA = am! = c' sin \d' . Also sz 
■=■ An' = c" cos \d'\ and sA = zn' = c" sin \d" . in which 





Fm. 14. 



Fig. 15. 



(d' -\- d") = D. The points ^ and a being determined on the 
ground, aa' may be computed and set off as before and the curve 
continued in full stations. A subchord at the end of the curve 
may be located by a similar process. 

30. Curve location by offsets from the long chord. (Fig. 16.) 
Consider at once the general case in which the curve commences 
with a subchord (curvature, d'), contains with one or more full 
cliords (curvature of each, D)^ and ends with a subchord with 
curvature d" . The numerical work consists in computing first 
AB^ then the various abscissae and ordinates. AB=2B sin ^Zi. 



Aa' = Aa! = c' cos ^{J — d'); 

Ah' = Aa' + a'b' = d cos \{A-d')-^' 00 cos \{A - 2d' - D) ; 

Ac' = Aa' + a'b' + b'c' = c' cos i(z/ - d') + 100 cos {{/I - 2d' - D) 

+ 100cosi(z/-2(f"-Z>); 
also 

-AB-Bc' =z2Bs\n^J- c" co%^{A - d"). 



Kii) 



§32. 



ALIGNMENT. 



29 



a'a=: a'a = c' sin |(z/ — d'); ") 

bb = a'a + mb= c sin 1{J - d')-\- 100 siu \{^ - 2d' - V); j 

c'c = bb - nb = c' sin 1{J - d')-{- 100 siu |(z/ - M - D) )■ (12) 

-lOOsin i(J-2(r'-Z>); \ 

also =c" s\u\{J — d"). J 



The above formulae are considerably simplified when the enrve 
begins and ends at even stations. When the curve is very long 
a regular law becomes very apparent in the 
formation of all terms between the first and last. 
There are too few terms in the above equations 
to show the law. 

31. Use and value of the above methods. The 
chief value of the above methods lies in the 
possibility of doing the work without a transit. 
The same principles are sometimes employed, 
even when a transit is used, when obstacles pre- 
vent the nse of the normal method (see § 32, c). 
If the terminal tangents have already been ac- 
curately determined, these methods are useful to 
locate points of the curve when rigid accuracy 
is not essential. Track foremen frequently use 
such methods to lay out unimportant sidings, 
especially when the engineer and his transit are not at hand. 
Location by tangential offsets (or by offsets from the long chord) 
is to be preferred when the curve is flat (i.e., has a small central 
angle ^) and there is no obstruction along the tangent, or long 
chord. Location by middle ordinates may be employed regard- 
less of the leno;th of the curve, and in cases when both the 




Fig. 16, 



tangents 



and the long chord are obstructed. The above 



methods are but samples of a large number of similar methods 
which have been devised. The choice of the particular 
method to be adopted nmst be determined by the local con- 
ditions. 

32. Obstacles to location. In this section will be given only 
a few of the principles involved in this class of problems, with 
illustrations. The engineer must decide in each case, which is 



30 



JRAILROAD CONSTRUCTION. 



32. 



tlie best metliod to use, and it is frequently advisable to devise a 
special solution for some particular case. 

a. When the vertex is inaccessible. As shown in § 26, it is 
not absolutely essential that the vertex of a curve should be 
located on the ground. But it is xery evident that the angle 
between the terminal tangents is determined with far less prob- 
able error if it is measured by a single measurement at tlie ver- 
tex rather than as the result of numerous angle measurements 
along the curve, involving several positions of the transit 
and comparatively short sights. Sometimes the location of the 
tangents is already determined on the ground (as by hi and am, 
Fig 17), and it is required to join the tangents by a curve of 
given radius. Method. Measure ab and the angles Vba and 
ha V. A is the sum of these angles. The distances h Fand a Y 
are computable from the above data. Given A and R, the tan- 





FiG. 18. 



gent distances are computable, and then Bb and aA are found 
by subtracting h V and a V from the tangent distances. The 
curve may then be run from A, and the work may be checked 
by noting whether the curve as run ends at £ — previously lo- 
cated from h. 

b. When the point of curve (or point of tangency) is inacces- 
sible. At some distance {As, Fig. IS) an unobstructed line pn 



§ 33. ALIGNMENT. 31 

may be run parallel with A V. nv = jpy = As =z Ji vers a. 



vers (Y = As ^ U. ns =j)s = It sin 



a. 



At ?/, which is at a distance jps back from the computed posi- 
tion of ^i, make an oifset sA to 7>>. Ilun i)ii parallel to the 
tano-ent. A tansjent to the curve at n makes an anoxic of « with 
np. From n the curve is run in as usual. 

. If the point of tangencj is obstructed, a similar process, 
somewhat reversed, may be used. /5 is that portion of A still 
to be laid off when m is reached, tin =^ tl =^ It sin /?. iriz = 
tB = lx =1 B vers ft. 

c. When the central part of the curve is obstructed. a is 
the central angle between two points of the curve between which 
a chord may be run. a may equal any angle, but it is prefer- 
able that a should be a multiple of Z^, the degree of curve, and 
that the points vi and n should be on even stations. m)i = 
2/t sin iot, A point s may be located 
by an oifset hs from the chord tnn by a ^^"""T^-^.^ 
similar method to that outlined in § 30. / tjS^^^^-,^ 

The device of introducin£i^ the dotted / / \'^o^ 

curve ???./? havino; the same radius of cur- / / \i'v^ 

vature as the other, although neither / / \\ 

necessary nor advisable in the case shown / / ^' 

in Fig. 19, is sometimes the best method a5^_, — - — """" 

of surveying around an obstacle. The 

oifset from any point on the dotted curve 

to the corresponding point on the true ^^^^' ^^• 

curve is twice the "ordinate to the long chord," as computed 

in § 30. 

33. Modifications of location. The following methods may 
be used in allowing for the discrepancies between the " paper 
location " based on a more or less rough preliminary survey and 
the more accurate instrumental location. (See § 15.) They are 
also frequently used in locating new parallel tracks and modify- 
ing old tracks. 



32 



RAILROAD CONSTRUCTION. 



33. 



a. To move the forward tangent parallel to itself a distance a?, 
the point of curve (^1) remaining fixed. (Fig. 20.) 



VV = -T 



V'h 



X 



sinAFF' sin ^ 



• • 



(13) 



AV = A V+ VV\ 
The triano^le BmB' is isosceles and Bin = B'm. 



R' - R^ O'O^mB^ 



B'r 



X 



vers BinB vers A 



.-. R'^R-\- 



X 



vers A 



. . . . (U) 



The solution is very similar in case the tangent is moved in- 
ward to V^'B^\ IS'ote that this method necessarily changes the 



Z^r-^ 




^-2^^^^ 


^^4-> 


/t^^ 


^^^^^ 




^^^^^-3: 




^\\ /i 


/ / r" 


\A 


/i/j/j 


\ 


0' 6 6" 









V' 

V 

V" 




Fig. 20. 



Fig. 21. 



radius. If the radius is not to be changed, the point of curve 
must be altered as follows : 

b. To move the forward tangent parallel to itself a distance x, 
the radius being unchanged. (Fig. 21.) In this case the whole 



§33. 



ALIGNMENT. 



33 



curve is moved bodily a distance 00' = A A' = W = BB', 
and moved parallel to the first tangent A V. 



BB' = -. 



B'n 



X 



sin iiBB' sin A 



= AA\ 



(15) 



c. To change the direction of the forward tangent at the point 
of tangency. (Fig. 22.) This problem involves a change («-) in 
the central angle and also requires a new radius. An error in the 
determination of the central angle furnishes an occasion for its 
use. 

^, J, a, A Vy and B Fare known. /}' = /} — a. 
Bs = B vers A. Bs = R' vers A' , 

vers A 



E' = E 



vers {A — ^) * 
As = E ^m A. A's z= E' sin A' , 



(16) 



.-. AA' = A's — As = E' sin A' — E sin A. . (17) 

The above solutions are given to illustrate a large class of 
problems which are constantly arising. All of the ordinary 





Fig. 22. 



Fig. 23. 



problems can be solved by the application of elementary ge- 
ometry and trigonometry. 



34 RAILROAD CONSTRUCTION. % 34. 

34. Limitations in location. It may be required to run a 
curve that shall join two given tangents and also pass through a 
given point. The point (P, Fig. 23) is assumed to be determined 
by its distance ( VP) from the vertex and by the angle A VP 

= p. 

It is required to determine the radius {E) and the tangent 

distance (^ F). A is known. 
PVG = i(180° - J) - /5 = 90° - (iJ + ^). 
PP' =2VP sin PVG ^2 VP cos (iJ + /?). 

PSV = U. .', SP = YP^""^ 



sin hA' 



AS = VSP X SP' = VSP{SF + PP')' 



= )/ FP4^r FP-g^ + 2 FP cos (iz^ + ^) 
sm t^L sm -^n 

__ Yp i/ sin' /g 2 sin yg cos (jz/ + jS) 



sin' \A sin ^-z/ 

sni \/i 

AV^AS^SY 

=_-^rsin (i// +/i) + |/sin' /3 + 2 sin ^ sin i^/ cos (i^ +/i)]. (18) 
sin ^i^ 

P:= ^FcOtiZl. 

In the special case in which P is on the median line F, 
y5 = 90° - iJ, and (Jz^ + /?)== 90°. Eq. (IS) then reduces to 

VP 

AV= ^-T^(l + cos i J) = FP cot iJ, 
sm 2 '^ 

as mi^ht have been immediately derived from Eq. (8). 



§ 35. ALIGNMENT. 35 

111 case the point P is given by the offset PK and by tlie 
distance YK^ the triangle PKV \\\a^\ be readily solved, giving 
the distance YP and the angle /?, and the remainder of the 
solution will be as above. 

35. Determination of the curvature of existing track, (a) Vs'niy 
a trcuisit. Set up the transit at any point in the center of the 
track. Measure in each direction 100 feet to points also in the 
center of the track. Sight on one point with the plates at 0°. 
Plunge the telescope and sight at the other point. The angle 
between the chords" equals the degree of curvature. 

(b) Using a tape and string. Stretch a string (say 50 feet 
long) between two points on the inside of the head of the outer 
rail. Measure the ordinate [x) between the middle of the string 
and the head of the rail. Then 

_, chord' . ^ ^ 

^ = — g^ (very nearly) (19) 

For, in Fig. 24, since the triangles AGE and ADC are 
similar, AO : AE : : AP : PJC or P = lAP" ~ x. When, 
as is usual, the arc is very short compared with 
the radius, AD = -J^-^? very nearly. Making 
this sul)stitution we have Eq. (19). With a 
chord of 50 feet and a 10° curve, the resulting 
difference in x is .0025 of an inch — far within 
the possible accuracy of such a method. The 
above method gives the radius of the inner head ^^^- ~^- 

of the outer rail. It should be diminished by %g for the radius 
of the center of the track. With easy curvature, however, this 
will not affect the result by more than one or two tenths of one 
per cent. 

The inversion of this formula gives the required middle or- 
dinate for a rail on a given curve. For example, the middle 
ordinate of a 30-foot rail, bent for a 6° curve, is 

a; = 900 -^ (8 X 955) = .118 foot = 1.1 inches. 




36 BAILBOAD CONSTRUCTION. § 36. 

Another much used rule is to require the foreman to have a 
string, knotted at the centre, of such length that the middle or- 
dinate, measured in inches, equals the degree of curve. To- 
find that length, substitute (in eq. (19)) 5730 -^- D for H and 
2> -=- 12 for X. Solving for chords we obtain chord =61.8 feet. 
The rule is not theoretically exact, but, considering the uncertain 
stretching of the string, the error is insignificant. In fact, the 
distance usually given is 62 feet, which is close enough for all 
purposes for which such a method should be used. 

36. Problems. A systematic method of setting down the 
solution of a problem simplifies the work. Logarithms should 
always be used, and all the work should be so set down that a 
revision of the work to find a supposed error may be readily 
done. The value of such systematic work M^ill become more 
apparent as the problems become more complicated. The twa 
solutions given below will illustrate such work. 

a. Given a 3° curve beginning at Sta. 27+60 and running 
to Sta. 32 -J- 45. Compute the ordinates and offsets used in 
locating the curve by tangential offsets. 

h. With the same data as above, compute the distances to 
locate the curve by offsets from the long chord. 

c. Assume that in Fig. 17 ab is measured as 217.6 feet,, 
the angle ah F= 17° 42', and the angle haV = 21° 14'. Join 
the tangents by a 4° 30' curve. Determine hB and aA, 

d. Assume that in a case similar to Fig. 18 it was noted 
that a distance {As) equal to 12 feet would clear the building. 
Assume that A = 38° 20' and that !> = 4° 40'. Eequired tiie 
value of a and the position of n. Solution : 

YQY^ a z=z As -^ R ^5=12 log = 1.07918 

R (for 4° 40' curve) log = 3.08923 

0^= 8° 01 ' log vers a = 7.98994 

m = B sin a log sin a = 9.14445 

log R = 3.08923 
71^ = 171.27 log = 2.2336^ 



§ 37. ALIGNMENT. 37 

e. Assume that the forward tangent of a 3° 2(V curve 
havinir a central anii^le of 1G° 50' must be moved 3.62 feet 
inward^ witliout altering the P. C. Required the cliange in 
radius. 

f. Given two tangents making an angle of 36° 18'. It is 
required to pass a curve through a point 93.2 feet from the 
vertex, the line from the vertex to the point making an angle 
of 42° 21' with the tangent. Required the radius and tangent 
distance. Solution: Applying eq. (18), we have 

2 

/? = 42° 21' 

ij = 18° 09' 
(ij + /?) = 60° 30' 

.20667 

logsinV = 9.656S8 .45382 

2| 9.81987 .66049 

9.90993 .81271 

nat sin 60° 30'= .8703 





log 


=z 


0.30103 


log 


sin 


= 


9.82844 


log 


sin 


= 


9.49346 


log 


cos 


^^ 


9.69234 
9.31527 



1.6836 


log= 0.22610 


YP= 93.2 


]og= 1.9694i 




2. 19551 




log sin iz/ =: 9.49346 


tang. dist. AY = 503.56 


log= 2.70205 




log cot iJ = 10.48437 


7?= 1536.1 


3.18642 



1) =-- 3° 44' 

COMPOUND CURVES. 

37. Nature and use. Compound curves are formed by a 
succession of two or more simple curves of different curvature. 
The curves must have a common tangent at the ])oint of com- 
pound curvature {P.C.C.). In mountainous regions there is 
frequently a necessity for compound curves having several 
changes of curvature. Such curves may be located separately 
as a succession of simple curves, *but a combination of two 



38 



RAILROAD CONSTRUCTION. 



3a 



simple curves has special properties wliicli are worth investigat- 
ing and utilizing. In the following demonstrations H^ always 
represents the longer radius and B^ the shorter^ no matter 
which succeeds the other. T^ is the tangent adjacent to the 
curve of shorter radius (A*,), and is invariably the shorter tan- 
o-ent. A^ is the central angle of the curve of radius i?j , but it 
may be greater or less than z/,. 

38. Mutual relations of the parts of a compound curve havings 
two branches. In .Fig. 25, ^6^ and CB are the two branches of 




Fig 25. 

the compound curve having radii of B^ and B^ and central 
ano-les of z/, and z/,. Produce the arc AC to n so that 
Ao{ii = A. The chord Cn produced must intersect B. The 
line ^16', parallel to CO^ , will intersect BO^ so that Bs = sn 
= (9^(9j = 7?^ — B^. Draw Am perpendicular to O^n. It will 
be parallel to lik. 

Br = S7i vers Bs7i = (7?, — B,) vers A^ ; 

m7i = AO^ vers A0^7i = R^ vers A ; 
Ak =^AV sin A Vh = T, sin A ; 
Ale = J nil = mn -\- nli = 7/171 -f- Br. 
,'. T^ sin A^B, vers A-\-{B, — B,) vers A^. . (20) 



§ 38. ALIGNMENT. 39 

Similarly it may be shown that 

7; sin A ^ R^ vers A — {B, — B,) vers A,. . (21) 

The mutual relations of the elements of compound curves 
may be solved by these two equations. For example, assume 
the tangents as fixed {/} therefore known) and that a curve of 
given radius Ii^ shall start from a given point at a distance T^ 
from the vertex, and that the curve shall continue through a 
given angle ^,. Required the other parts of the curve. From 
Eq. (20) we have 

T, sin J — i?i vers A 



i?, - 7?, = 



vers ^2 



T, sin /I- B^ vers /I 

„•. It., = li -] J-. -rr . . . (22) 

* ' vers (// — -^j ^ ^ 

T^ may then be obtained from Eq. (21). 

As another problem, given the location of the two tangents, 
with the two tangent distances (thereby locating the PC and 
PT)^ and the central angle of each curve ; required the two 
radii. Solving Eq. (20) for R^ , we have 

T. sin A — R„ vers J„ 



vers A — vers A^ 



Similarly from Eq. (21) we may derive 

„ _ J!, sin A — R^ (vers A — vers z/,) 
vers ^1 

Equating these, reducing, and solving for R^ , we have 

T, sin A vers z/, — T^ sin A (vers A — vers z^,) 
'''^ ~ vers A^ vers ^, — (vers A —vers ^,)(vers A — vers A^' ^^ 

Althouorh the various elements mav be chosen as above with 
considerable freedom, there are limitations. For example, in 
Eq. (22), since R^ is always greater than R^ , the term to be 
added to R^ must be essentially positive — i.e., T^ sin A must be 



40 



RAILROAD CONSTRUCTION. 



§«^9. 



VGrs ^ 
greater than 7?, vers /I. This means that 7T > IL —. — ;r* 

or that T^ > ^^ tan |-z/, or that T, is greater than the corre- 
sponding tangent on a simple curve. Similarly it may be 
shown that T^ is less than i?, tan ^^ or less than the correspond- 
ing tangent on a simj^le curve. E^evertheless T^ is always 
greater than T^, In the limiting case when /^ = -^i , T, = T 
and z/^ — ^1- 

39. Modifications of location. Some of these modifications 
may be solved by the methods used for simple curves. For 
example : 

a. It is desired to move the tangent VB, Fig. 26, parallel to 
itself to V^B\ Run a new curve from the P. C. O. which shall 
reach the new tangent at B\ where the chord of the old curve 





Fig. 26. 



Fig. 27. 



intersects the new tangent. The solution is almost identical with 
that in § 33, c^. 

b. Assume that it is desired to change the forward tangent 
(as above) but to retain the same radius. In Fig. 27 

(7?2 — ^i) cos J, = O^n ; 
(7?, - B,) cos /},' = 0:n'. 
X — O^n — O^n' = {R^ — P,)(cos z/^ — cos z//). 

X 
cos /^/ = cos Z/^ — n __ p ' . . . (24) 



§39. 



ALIGNMENT. 



41 



The P. C. C. is moved hackward along the sliarper curve an 
angular distance of ^/ — ^^ — z^i — /^/. 

In case the tangent is moved inward rather than outward, 
the solution will apply bj transposing A^ and ^/. Then we 
will have 



cos 



A' = 



cos ^. + ' p _ p - . . . (25) 



Tlie P. C. C. is then moved for- 
ward. 

c. Assume the same case as (b) ex- 
cept that the larger radius comes first 
and that the tangent adjacent to the 
smaller radius is moved. In Fisr. 28 

(i?2 — R,) cos J, = 0,71 ; 

(i?, - B,) cos zf/ = 0;n', 




'' "n^-^ 



Fig. 28. 



X = 0/n' - 0^71 = (P, - i?,)(cos z// _ cos ^,). 

cos /I/ = COS A, + p^^_^ > . . . (26) 



The P. C. C. is moved forward along the easier curve an 
angular distance of ^/ —. /i^ = /l^ — A^, 

In case the tangent is moved inward, transpose as before and 
we have 



cos Al = cos A^ — 



X 



p.-pr 



. . (27) 



The P. C. C. is moved hachioard. 

d. Assume that the radius of one curve is to be altered witli- 
out changing either tangent. Assume conditions as in Fig. 29. 
For the diagrannnatic solution assume that P, is to be in- 



42 



RAILROAD CONSTRUCTION. 



39. 



creased by O^S, Then, since ^/ must pass through 0, and ex- 
tend beyond 0, a distance 0,S, the locus of the new center 
must lie on the arc drawn about 0, as center and with OS as 

radius. The locus of 6*/ is also given 
by a line O^'j? parallel to ^ F and at a 
distance of ^/ (equal to S ... P. CO.) 
from it. The new center is therefore 
at the intersection 0^\ An arc with 
radius i?/ will therefore be tangent at 
Jj' and tangent to the old curve ^^r^- 
chiced at new P.C.C. Draw O^n 
perpendicular to O^B. With 0^ as 
center draw the arc 0,m, and with 
0^ as center draw the arc 0{in' . 
mB = m' B' = B,. .-. mn = m'n' = 




\s/U 



Fig. 29. 
(i?/ - B,) vers J/ = {B, - B,) vers /J,. 



•. vers /i: = ^, ^ vers J, 

{It^ — It,) 



. . (28) 



0,71= (B,-B,) sin/},', 



0,n'={B:-B,)sm z//. 



BB' = 0,n'-0,n - {B:-B,) sin zJ/- {B~B) sin z^,. (29) 



This problem may be further modified by assuming that the 
radius of the curve is decreased rather than increased, or that the 
smaller radius follows the larger. The solution is similar and 
is suggested as a profitable exercise. 

It might also be assumed that, instead of making a given 
change in the radius ^,, a given change BB' is to be made, 
z?/ and ^/ are required. EHminate B^' from Eqs. 28 and 29 
and solve the resulting equation for z^/. Then determine B^' by 
a suitable inversion of either Eq. 28 or 29. 



§ 41. ALIGNMENT. 43 

As in §§ 32 and 33, the above problems are but a few, 
although perhaps the most common, of the problems tlie 
engineer may meet with in compound curves. All of tlie 
ordinary problems may be solved by these and simihir 
methods. 

40. Problems, a. Assume that the two tangents of a com- 
pound curve are to be 348 feet and 624 feet, and that ^, = 
22° 16' and ^, = 28° 20'. Kequired the radii. 

[Ans. ^,:= 326.92; 7?, = 1574.85.] 

h. A line crosses a valley by a compound curve which is first 
a 6° curve for 46° 30' and then a 9° 30' curve for 84° 16'. It is 
afterward decided that the last tangent should be 6 feet farther up 
the hill. What are the required changes ? {Note. The second 
tangent is evidently moved outward. The solution corresponds 
to that in the first part of § 39, c. The P. C. C. is moved forward 
16.39 feet. If it is desired to know how far the P. T. is moved 
in the direction of the tangent (i.e., the projection of JSJ3\ Fig. 
28, on V B), it may be found by observing that it is equal to 
7in' = (7?2 — ^,)(sin A, — sinz//). In this case it equals 0.65 
foot, which is very small because A^ is nearly 90°. The value 
of z/^ (^^° ^^0 is not used, since the solution is independent of 
the value of A^. The student should learn to recognize which 
quantities are mutually related and therefore essential to a solu- 
tion, and which are independent and non-essential.] 



TRANSITION CURVES. 

41. Superelevation of the outer rail on curves. When a mass 
is moved in a circular path it requires a centripetal force to keep 
it moving in that path. By the principles of mechanics we 
know that this force equals Gv' -r- gli^ in which G is the weight, 
V the velocity in feet per second, g the acceleration of gravity 
in feet per second in a second, and R the radius of curvature. 
If the two rails of a curved track were laid on a level (trans- 
versely), this centripetal force could only be furnished by the 



44 



RAILROAD CONSTRUCTION. 



41. 




pressure of the wlieel-flanges against tlie rails. As tliis is very 
objectionable, the outer rail is elevated so that the reaction of 

the rails against the wheels shall contain 
a horizontal component equal to the re- 
cpiired centripetal force. In Fig. 30, if 
oh represents the reaction, oc will repre- 
sent the weight G, and ao will represent 
___^_-l--— -"[^ the required centripetal force. From 
similar triangles we may write S7i : sm : : 
ao\oc. Call ^=32.17. Call B = 
5730 -^ Z), which is sufficiently accurate 

for this purpose (see § 19). Call v= 5280 F-^ 3600, in 
whieh T^is the velocity in miles per hour, mn is the distance 
between rail centers, which, for an SO-lb. rail and standard 
gauge, is 4.916 feet, sin is slightly less than this. As an 
average value we may call it 4.900, which is its exact value 
when the superelevation is 4i inches. Calling sn = <?, we have 



Fig. 30. 



sm 



ao 
oc 

e 



= 4.9 



Gv' 1 



4.9 X 5280' F'i> 



gR G 32.17 X 3600' X 5730 
.0000572 F'Z> 



(30) 



It should be noticed that, according to this formula, the 
required superelevation varies as the sqttare of the velocity, 
which means that a change of velocity of only 10,^ would call 
for a change of superelevation of 21^. Since the velocities of 
trains over any road are extremely variable, it is impossible to 
adopt any superelevation which will fit all velocities even 
approximately. The above fact also shows why any over- 
refinement in the calculations is useless and why the above 
approximations, which are really small, are amply justifiable. 
For example, the above formula contains the approximation that 
i? = 5730 -f- Z^. In the extreme case of a 10° curve the error 
involved would be about Ifo, A change of about i of I'fo in 



§42. 



ALIGNMENT. 



46 



the velocity, or say from 40 to 40.2 miles per hour, would mean 
as much. The error in e due to tlie assumed constant value 
of s?n is never more than a very small fraction of Ifc. Tlie 
rail-laying is not done closer than this. The following tabular 
form is based on Eq. 30 : 

SUPERELEVATION OF THE OUTER RAIL (IN FEET) FOR VARIOUS VELOCI- 
TIES AND DEGREES OF CURVATURE. 



Velocity 

ill 

Miles 

per 

Hour. 


Degree of Curve. 


1° 


o& 


3° 

.15 
.27 
.43 


4° 

.20 

.37 


5° 


6° 


7° 


8° 


9» 


10° 


30 
40 
50 
60 


.05 
.09 
.14 
.20 


.10 
.18 
.29 
.41 


.26 
.46 


.31 


.36 


.41 


.46 


1.51 


i .55 
.86 


.64 


.73 


.82 


1 .57 

.82 


.71 


1 .62 



42. Practical rules for superelevation. A much used rule 
for superelevation is to ''elevate one half an inch for each 
degree of curvature." The rule is rational in that e in Eq. 30 
varies directly as I), The above rule therefore agrees with 
Eq. 30 when Fis about 27 miles per hour. However applica- 
ble the rule may have been in the days of low velocities, the 
elevation thus computed is too small now. 

Another (and better) rule is to "elevate for the speed of the 
fastest trains." This rule is further justified by the fact that a 
four-wheeled truck, having two parallel axles, will always tend 
to run to the outer rail and will require considerable flange 
pressure to guide it along the curve. The effect of an excess of 
superelevation on the slower trains will only be to relieve this 
flange pressure somewhat. This rule is coupled with the limita- 
tioirthat the elevation should never exceed a limit of six inches 
—sometimes eight inches. This limitation implies that locomo- 
tive engineers must reduce the speed of fast trains around sharp 
curves until the speed does not exceed that for which the actual 
superelevation used is suitable. The heavy line in tlie tabular 
form (§ 41) shows the six-inch limitation. 



46 



BAILTtOAD CONSTRUCTION, 



§48. 



Some roads furnish their track foremen with a Hst of the 
superelevations to be used on each curve in their sections. 
This method has the advantage that each location may be 
separately studied, and the proper velocity, as affected by local 
conditions {e.g.^ proximity to a stopping-place for all trains), 
may be determined and applied. 

Another method is to allow the foremen to determine the 
superelevation for each curve by a simple measurement taken 
at the curve. The rule is developed as follows : By an inversion 
of Eq. 19 we have 



X 



= chord' -^ 8B (31) 



Putting X equal to ^ in Eq. 30 and solving for ^'chord,^^ we 
have 



chord' = .0000572 V'DSE 

= 2.621 F\ 
chord = 1.62 F. . 



(32> 



To apply the rule, assume that 50 miles per hour is fixed as 
the velocity from which the superelevation is to be computed. 
Then 1.62 F== 1,62 X 50 = 81 feet, which is the distance given 
to the trackmen. Stretch a tape (or even a string) with a 
length of 81 feet between two points on the inside head of the 
outer rail or the outer head of the inner rail. The ordinate at 
the middle point then equals the superelevation. The values 
of this chord length for varying velocities are given in the 
accompanying tabular form. 



Velocity in miles per hour 

Chord length in feet 


•20 
32.4 


25 30 
40.5 48.6 


35 
56.7 


40 
64.8 


45 
72.9 


50 
81.0 


55 
89.1 


60 
97.2 



43. Transition from level to inclined track. On curves the 
track is inclined transversely; on tangents it is level. The 
transition from one condition to the other must be made gradu- 



§ 45. ALIGNMENT. 



47 



ally. If there is no transition curve, there must be either in- 
clined track on the tangent or insufficiently inclined track on the 
curve or both. Sometimes the full superelevation is continued 
through the total length of the curve and the " run- oft" " 
(having a length of 100 to 200 feet) is located entirely on the 
tangents at each end. In other practice it is located partly on 
the tangent and partly on the curve. Whatever the method, 
the superelevation is correct at only one point of the run-olf. 
At all other points it is too great or too small. This (and other 
causes) produces objectionable lurches and resistances when 
entering and leaving curves. The object of transition curves is 
to obviate these resistances. 

44. Fundamental principle of transition curves. If a curve 
has variable curvature, beginning at the tangent with a curve of 
infinite radius, and the curvature gradually sharpens uiitil it 
equals the curvature of the required simple curve and there 
becomes tangent to it, the superelevation of such a transition 
curve may begin at zero at the tangent, gradually increase to 
the required superelevation for the simple curve, and yet have 
at every point the superelevation required by the curvature at 
that point. Since in Eq. (30) e is directly proportional to />, 
the required curve must be one in which the degree of curve 
increases directly as the distance along the curve. The mathe- 
matical development of such a curve is quite complicated. It 
has, however, been developed, and tables have been computed for 
its use, by Prof. C. L. Crandall. The following method has 
the advantage of great simplicity, while its agreement w^itli the 
true transition curve is as close as need be, as will be shown. 

45. Multiform compound curves. If the transition curve 
commences with a very flat curve and at regular even chord 
lengths compounds into a curve of sharper curvature until the 
desired curvature is reached, the increase in curvature at each 
chord point being uniform, it is plain that such a curve is a 
close approximation to the true spiral, especially since the rails 
as laid will gradually change their curvature rather than main- 
tain a uniform curvature throughout each chord loTi£rtli and 



48 RAILROAD CONSTRUCTION. § 46. 

then abruptly change the curvature at the chord points. Such 
a curve, as actually laid^ will be a much closer approximation 
to the true curve than the multiform compound curve by which 
it is set out. There will actually be a gradual increase in 
curvature which increases directly as the length of the curve. 

46. Required length of spiral. The required length of spiral 
evidently depends on the amount of superelevation to be 
gained, and also depends somewhat on the speed. If the spiral 
is laid off in 25-foot chord lengths, with the first chord subtend- 
ing a 1° curve, the second a 2° curve, etc., the fifth chord will 
subtend a 5° curve, and the increase from this last chord to a 
6° curve is the same as the uniform increase of curvature 
between the chords. The same spiral extended would run on 
to a 12° curve in (12 - 1)25 = 275 feet. The last chord of a 
spiral should have a smaller degree of curvature than the simple 
curve to which it is joined. If the curves are very sharp, such 
as are used in street work and even in suburban trolley work, 
an increase in degree of curvature of 1° per 25 feet will not be 
sufticiently rapid, as such a rate would require too long curves. 
2°, 10°, or even 20° increase per 25 feet may be necessary, but 
then the chords should be reduced to 5 feet. Such a rapid rate 
of increase is justified by the necessary reduction in speed. On 
the other hand, very high speed will make a lower rate of 
increase desirable, and therefore a spiral whose degree of curva- 
ture increases only 0° 30' per 25 feet may be used. Such a 
spiral would require a length of 375 feet to run on to an 8° 
curve, which is inconveniently long, but it might be used to 
run on to a 4° curve, where its length would be only 175 feet. 
Three spirals have been developed in Table lY, each with chords 
of 25 feet, the i*ate of increase in the degree of curvature being 
0° 30', 1° and 2° per chord. One of these will be suitable for 
any curvature found on ordinary steam-railroads. 

47. To find the ordinates of a l°-per-25-feet spiral. Since the 
first chord subtends a 1° curve, its central angle is 0° 15' and 
the angle aQY (Fig. 31) is 7' 30". The tangent at a makes an 
angle of 15' with VQ. The angle between the chord ha and 



§48. 



ALIGNMENT. 



49 



die tangent at a is J(30') ^ 15', and the angle hah"= 4(30') + 15' 
= 30'. Similarly the angle chc" = i(45') + 3U' + 15' ^ OT' 3U'' 
= 1° 07' 30", and the angle dcd" is 2° 0'. The ordinate aa 
= 25 sin 7' 30", and Qa = 25 cos 7' 30". Qh' = Qa! + aV 
^ Qa: 4- aV = 25 (cos 7' 30" + cos 30'). hb' = W + ^^/' 
= 25 (sin 7' 30" + sin 30'). Similarly the ordinates of c, d, 
etc., may be obtahied. 




Fig. 31. 




FrG. 32. 



48. To find the deflections from any point of the spiral. 

aQV=7' 30". Tan hQ V =^ hh' -^ QV ; tan cQV = cc' -^ Qc ; 
etc. Thus we are enabled to find the deflection angles from 
the tangent at Q to any point of the spiral. 

The tangent to the curve at c (Fig. 32) makes an angle of 
1° 30' with Q V, or cm F = 1° 30'. Qcm = cm V - cQm. The 



50 



RAILROAD COI^STRUCTION. 



48. 



value of cQm is known from previous work. The deflection 
from c to Q then becomes known. 

acm = cmV — cap = ciiiV — caq — qap, caq is the deflec- 
tion angle to c from the tangent at a and will have been 
previously computed numerically, qaj? = 15'. acm therefore 
becomes known. 

hcm^iofW = 22' 30"; 

den = ioi 60' = 30'. 

ecn = ecd"— ncd'\ ncd" = cmV^ tan ecd" = {ee'— d"d')-r- c'e\ 
all of which are known from the previous work. 

By this method the deflections from the tangent at any 




Fig. 33. 

point of the curve to any other point are determinable. These 
values are compiled in Table lY. The corresponding values 
of these angles when the increase in the degree of curvature per 
chord length is 30', and when it is 2°, are also given in 
Table lY. 



§ 49. ALIGNMHNT. 51 

49. Connection of spiral with circular curve and with tangent. 
See Fig. 3o.''' Let A V and I^ V be the tangents to be cunnected 
bj a D° curve, having a suitable spiral at each end. If no 
spirals were to be used, the problem would be solved as in 
simple curves giving the curve AMB. Introducing the spiral 
has the effect of throwing the curve away from the vertex a 
distance MM' and reducing the central angle of the D^ curve 
by 20. Continuing the curve beyond Z and Z' \o A' and B\ 
we will have AA' = BB' = MM'. ZK — the x ordinate and 
is therefore known. Call MM' — m. A'N =^ x — R vers 0. 
Then 

n.^^^, 4 4, ^'^ X — R vers ,^^^ 

m = MM' = AA' = —: == — , . . . (33) 

cos ^A cos "l"^ ^ ^ 

iTJL = AA ' sin l^ = (x — R vers 0) tan i^. 
VQ = QK- KN-\-NA + AV 

— ij — R sin + (,^; — 7? vers 0) tan \A ^ R tan \A 

— y — R sin + a; tan 4-^ + i^ cos tan \A, . (34) 

When A 'N has already been computed, it may be more con- 
venient to write 

VQ=zy^R (tan ^A - sin 0) + A'N tan ^A. (35) 
V2r ^ VM+MM' 

= i? exsec i^ A —. r^p. . . (oo) 

^ ' cos ^A cos ^^ ^ ^ 

= ?/ — i? sin + (a? — ^ ^^ers 0) tan ^A. (37) 

Example. To join two tangents making an angle of 34° :2(>' 
by a 5° 40' curve and suitable spirals. Use l°-per-25-feet 

* The student should at once appreciate the fact of the necessary distor- 
tion of the figure. The distance MM' in Fig. 33 is perhaps 100 times its real 
proportional value. 



52 



RAILROAD CONSTRUCTION. 



§50. 



spirals with five chords. Then = 3° 45^ x = 2.999, ^J 
= 17° 10', and y = 124.942. 



(Eq. 33) 



B 

vers 



312.471 

AQ = 59.042 



AF 



3.00497 
7.33063 













2.166 




0.33560 










X = 


2.999 














A'N = 


0.833 


cos ^A 


9.92064 
9.98021 






771 = 


= Mir = 


AA' =^ 


0.872 




9.94043 


(Eq. 


36) 










R 


3.00497 












exsec ^A 


8.66863 










V3f = 


: 47.164 




1.67365 










on = 


0.872 








35) 


y = 


= 124.942 


V2r = 

nat. 


48.036 


= .30891 




(Eq. 


tan Jz/ - 












nat. 


sin = 


= .06540 
.24351 
R 


9.3865! 
3.00497 








246.314 


[See 


above^ 


A'N- 


2.39148 




9.92064 














tan ^A 


9.48984 






VQ = 


0.257 






Aj}^ 


9.41048 




= 371.513 




(Eq. 


37) 










R 

tan \A 


3.00497 
9.48984 



2.49481 



50. Field-work. When the spiral is designed during the 
original location, tlie tangent distance VQ should be computed 
and the point Q located. It is hardly necessary to locate all of 
the points of the spiral until the track is to be laid. The 
extremities should be located, and as there will usually be one 
and perhaps two full station points on the spiral, these should 



§ 51. ALIGNMENT. 53 

also be located. Z may be located by setting off QK — y and 
KZ ^= a', or else by the tabular deflection for Z from Q and the 
distance ZQ, which is the long chord. Setting up the instru- 
ment at Z and sighting back at Q with the proper deflection, the 
tangent at Z may be found and the circular curve located as 
usual, its central angle being ^ — 20. A snnilar operation will 
locate Q' from Z' . 

To locate points on the spiral. Set up at Q, with the plates 
reading 0" when the telescope sights along VQ. Set oft" from 
Q the deflections given in Table IV for the instrument at Q, 
usino" a chord length of 25 feet, the process being like the 
method for simple curves except that the deflections are irregu- 
lar. If a full station-point occurs within the spiral, interpolate 
between the deflections for the adjacent spiral- points. For ex- 
ample, a spiral begins at Sta. 5G + 15. Sta. 57 comes 10 feet 
beyond the third spiral point. The deflection for the third point 
is 35' 0"; for the fourth it is 56' 15". |f of the difference 
(21' 15") is 8' 30" ; the deflection for Sta. 57 is therefore 43' 30". 
This method is not theoretically accurate, but the error is small. 
Arriving at z, the forward alignment may be obtained by sight- 
ing back at Q (or at any other point) with the given deflection 
for that point from the station occupied. Then when the plates 
read 0° the telescope will be tangent to tlie spiral and to the 
succeeding curve. All rear points should be checked from z. 
If it is necessary to occupy an intermediate station, use the de- 
flections given for that station, orienting as just explained for 5, 
checking the back points and locating all forward points up to z 
if possible. 

After the center curve has been located and z' is reached, the 
other spiral must be located but in reverse order, i.e., the sharp 
curvature of the spiral is at z' and the curvature decreases toward 

Q'- 

51. To replace a simple curve by a curve with spirals. This 
may be done by the method of § 49, but it involves shifting the 
whole track a distance m^ which in the given example equals 
0.87 foot. Besides this the track is appreciably shortened. 



54 



RAILROAD CONSTRUCTION. 



§51. 



which would require rail-cutting. But the track may be kept at 
j^racticallj the same length and the lateral deviation from the 
old track may be made very small by slightly sharpening tlie 
curvature of the old track, moving the new curve so that it is 
wholly or partially outside of the old curve, the remainder of it 
witli the spirals being inside of the old curve. It is found by 
experience that a decrease in radius of from Ifo to 5^ will answer 




o 



Fig. 34. 

the purpose. The larger the central angle the less the change. 
The solution is as indicated in Fig. 34. 

O'V = (9'ir sec 1^ 

= ^ ' cos sec ^^ -\- X sec ^^, 
m = MM' = 2fV-M'V 

= Eex8eci^ -{O'V - B') 

= B exsec J^ — B' cos sec ^^ — x sec J^ + B'. (38) 
AQ = QK- KN'-\-.¥V- YA 

— y—B' sin + (^' cos + x) tan ^^ — B tan ^^ 

=y—B' sin <P -\- B' cos tan ^A — {B — x) tan i^. (39) 



§51. 



ALIGNMENT. 



oa 



The length of the old curve from Qio Q' = '2.AQ '\- lOOy 

The length of the new curve from Qio Q' = "IL -\- 100 — 77-^, 

in Avhich L is the length of each spiral. 

Example. Suppose the old curve is a Y° 30' curve with a 
central angle of 38° 40'. As a trial, compute the relative length 
of a new 8° curve with spirals of seven chords. 0=7° 0' ; 
iJ = 19° 20' ; R (for the 7° 30' curve) = 76i.-189 ; jR' (for the 
8° curve) = 716.779; aj = 7.628. 

[Eq. 38] 



45.687 
R' = 716.779 



[Eq. 89] 



762.466 



762.037 
7n = 0.429 



< 53. 953 



8.084 
762.037 



R 

exsec 5 J 


2.88337 
8.77642 




1.65979 


R' 
cos (p 

sec iz/ 


2.85538 
9.99675 
0.02521 




2.8773+ 


X 

sec ^A 


0.88241 
0.02521 




0.90762 



y = 174.722 


87.353 
265.543 


R' 

sin (p 

R' 
cos (p 
tan I A 

R= 764.489 

x= 7.628 


2.85538 

9.08589 

1.94128 




2.85538 
9.99675 
9.54512 


249.606 


2.39725 








756.861 
tan ^J 


2.8790? 
9.54512 

2.42413 


424.328 
352.896 


352.896 




AQ= 71.432 





56 RAILROAD CONSTRUCTION. § 52. 

The length of the old curve from Q to Q' is 

100^ = lO^TX^ = 515.556 

2^^=2X71.433= 142.864 



New curve : 100^-=^ = ,,, 38.667-14.000 ^ ^^^ ^^^ 
JJ o.U 

2i: = 2 X 175 =350.000 



658.420 



658.333 658.333 
Difference iu length = 0.087 

Considering that this difference may be divided among 22 
joints (using 30-foot rails) no rail-cutting would be necessary. 
If the difference is too large, a slight variation in the value of 
the new radius R' will reduce the difference as much as neces- 
sary. A truer comparison of tlie lengths would be found by 
comparing the lengths of the arcs. 

52. Application of transition curves to compound curves. 
Since compound curves are only employed when the location is 
limited by local conditions, the elements of the compound curve 
should be determined (as in §§38 and 39) regardless of the 
transition curves, depending on the fact that the lateral shifting 
of the curve when transition curves are introduced is very 
small. If the limitations are very close, an estimated allowance 
may be made for them. 

Methods have been devised for inserting transition curves 
between the branches of a compound curve, but the device is 
complicated and usually needless, since when the train is once on 
a curve the wheels press against the outer rail steadily and a 
change in curvature will not produce a serious jar even though 
the superelevation is temporarily a little more or less than it 
should be. 

If the easier curve of the compound curve is less than 3° or 
4°, there may be no need for a transition curve off from that 
branch. This problem then has two cases according as transition 
curves are used at both ends or at one end only. 



§52. 



ALIGNMENT. 



57 



a. With transition curves at loth ends. Adopting the 
method of ^ 49, calling ^, = i^, we may compute m, = J/"J//. 
Similarly, calling A, = 4^, we may compute m, = 2IMJ. But 




Fig. 35. 

1// and MJ must be made to coincide. This may be done by 
moving the curve Z'3/"/ and its transition curve parallel to (/V 
a distance M/M,, and the other curve parallel to QV ii distance 
M'M,. In the triangle M,'MJf,\ the angle at 31,' = 00° - J„ 
tlie angle at MJ = 90° - ^,, and the angle at 21, = ^. 

sin(90°-Z/,) , .cos ^, 

Then M,'M, = M:M:-^^^^={^^-^^h)^^- 

sin(90°-^;) , /'OS -^, 

Similarly 2f:2f, = iZ/^^/Z-^g-Tj— =(^^-"^'^ih71J 



m 



58 BAILROAD CONSTRUCTION, % 53. 

b. With a transition curve on the sharper curve only. Com- 
pute 7?^, = MMl as before ; then move tlie curve Z^Ml parallel 
to Q'V Si distance of 



Jf/Jf, = m,5?^^ (41) 

sm ^ ^ 



The simple curve MA is moved parallel to VA a distance of 



3£3£, = 971!"-^ (42) 

sm ^ ^ 



If /J, and ^5 are both small, M/M^ and IfM^ may be more 
than mj, but the lateral deviation of the new curve from the old 
will always be less than ^n^. 

63. To replace a compound curve by a curve with spirals. 
The solution is somewhat analogous to that of § 51. Compute 
7n^ for the sharper branch of the curve, placing ^1 = ^^ in Eq. 
38. Since 7n^ and m^ for the two branches of the curve must 
be identical, a value for ^/ must be found which will satisfy 
the determined value of 7n^ = 7n^. Solving Eq. 38 for B', we 
obtain 

„ B vers i^ — m cos ^A — x ..^. 

cos — COS t^ ^ ^ 



Substituting in this equation the known value of 772/j (= m^ 
and calling R' = B^\ 7? = i?^, and A^ = ^A^ solve for j^,'. 
Obtain the value oi AQ for each branch of the curve separately 
by Eq. 39, and compare the lengths of the old and new lines. 

Example. Assume a compound curve with Z>, = 8°; Z^^ =r 4° ; 
^, = 36" and 4, = 32°. Use l°-per-25-feet spirals ; 0^ = 7° 0' ; 
02 = 1° 30'. Assume that the sharper curve is sliarpened from 
8° 0' to 8° 12'. 



§53. 

[Eq. 38] 



ALIGNMENT. 



169.209 
Ri' = 699.326 



868.535 



857.970 



9.429 



[Eq. 43] 



215.974 



nat. cos (p = .99966 
nat. cos /1 2 — .84805 



i?,' = 1424.54 [4°1'22'] 



[Eq. 39] 



2/, = 174.722 



85.226 



504.302 



679.024 
600.461 



515.235 

600.461 



^Q, = 78.563 



59 



exsec 36° 


2.85538 
9.37303 




2.22842 


11.' 

cos 01 

sec z/i 


2.84408 
9.99075 
0.09204 




2.93347 


3*1 

sec A I 


0.88241 
0.09204 



0.97445 



m\ 


867.399 
= 1.136 

217.700 
1.726 


^•2 


867. 

0. 
= 0, 

1 


399 

963 
.763 

.726 


vers 32° 

m, = 1.136 
cos 32° 






3.15615 

9.18175 

2.33785 




0.05538 
9.92842 




9.98380. 



2.33440 



.15161 


9.1807S 




3.15307 


sin 01 


2.84468 
9.08589 




1.93057 


cos 01 

tan \AIA, = 36°] 


2.84468 
9.99675 
9.86126 


R, = 716.779 

X, = 7.628 


2 . 70269 




709.151 
tan \A 


2.85074 
9.86126 



2.7120a 



60 

[Eq. 39] 



RAILROAD CONSTRUCTION. 



2/a = 74.994 



889.843 



964.837 
932.060 



37 290 



894.770 
932.060 



AQ-.^ 32.777 
For the length of the old track we have 



53. 



R,' 

sin 02 


3.15367 
8.41792 




1.57159 


R,' 

cos 0a 

tan iz/(J2 = 32°) 


3.15367 
9.99985 
9.79579 




2.94931 


i?a = 1432.69 

CCa = 0.76 




1431.93 
tan iz/ 


3.15592 
9.79579 



2.95171 



100 
100 



D, 


= "0 - 


= 450. 


^2 
i>2 


= 100 -° 


= 800. 




AQ, 


= 78.563 




AQ^ 


= 32.777 
1361.340 



For the length of the new track we have : 



100^i^'^ = 100-J?l= 353.659 



i>/ 



8°.20 



100^i^l^ = 10oS.= 758.140 



A 



4°. 023 



Spiral on 8° 12' curve 175.000 

" " 4° 01' 22" " 75. 



Length of new track = 1361.799 

" old " = 1361.340 



Excess in length of new track = 0.459 feet. 



§ 55. ALIGNMENT. 61 

Since the new track is slightly longer than the old, it shows 
that the new track runs too far outside tlie old track at the 
P.C.C. On the other hand the offset 7n is only 1.186. The 
maximum amount by which the new track comes inside of the 
old track at two points, presumably not far from Z' and Z, is 
vei-y dithcult to determine exactly. Since it is desirable that the 
maxinnini offsets (inside and outside) should be made as nearly 
equal as possible, this feature should not be sacrificed to an effort 
to make the two lines of precisely equal length so that the rails 
need not be cut. . Therefore, if it is found that the offsets inside 
the old track are nearly equal to m (1.136), the above figures 
should stand. Otherwise vi may be diminished (and the above 
excess in length of track diminished) by increasing R^ very 
slightly and making the necessary consequent changes. 



VERTICAL CURVES. 

54. Necessity for their use. AYhenever there is a change in 
the rate of grade, it is necessary to eliminate the angle that 
Avould be formed at the point of change and to connect the two 
grades by a curve. This is especially necessary at a sag be- 
tween two grades, since the shock caused by abruptly forcing 
an upward motion to a rapidly moving heavy train is very 
severe both to the track and to the rolling stock. 

55. Required length. Theoretically the length should de- 
pend on the change in the rate of grade, the greater change 
requiring a longer curve. The importance of this was greater 
in the days when link couplers were in universal use and the 
^' slack " in a long train was very great. Under such circum- 
stances, when a train was moving down a heavy grade the cars 
would crowd ahead against the engine. Reaching the sag, the 
engine would begin to pull out, rapidly taking out the slack. 
Six inches of slack on each car would amount to several feet on 
a long train, and the resulting jerk on the couplers, especially 
those near the rear of the train, has frequently resulted in 



62 RAILROAD CONSTRUCTION. § oQ. 

broken couplers or even derailments. A vertical curve will 
practically eliminate tins danger if the curve is made long 
enough, but the rapidly increasing adoption of close spring 
couplers and air-brakes, even for freight trains, is obviating the 
necessity for such very long curves. Two hundred feet may be 
considered sufficiently long for all ordinary changes of grade. 
Four hundred feet would probably suffice for the greatest 
change ever found in practice. 

56. Form of curve. In Fig. 36 assume that A and C, equi- 



FiG. 36. 

distant from B^ are the extremities of the vertical curve. Bi- 
sect AG ^i e\ draw Be and bisect it at h. Bisect AB and BC 
at h and I. The line kl will pass through h. A parabola may 
be drawn with its vertex at h which will be tangent to AB and 
BC 2ii A and B. It may readily be shown from the proper- 
ties of a parabola that if an ordinate be drawn at any point (as 
at n) we will have 



sn : eh (or JiB) ; : Am : Ae , 

, Aori" 
or sn = eri—j~Y (44) 

Since the elevation of any point along AB or BO is readily 
determinable, the elevation of any point on the curve may be 
computed by adding the correction sn. 

57. Numerical example. Assume that B is located at Sta. 
16 + 20; that the curve is to be 200 feet long; that the grade 
of AB is — 0.8^, and of B0-{- 1.2^; also that the elevation 
of B above the datum plane is 162.6. Then the elevation of 
the various points is as follows: A, 163.4; 0, 163.8; 6, 



§ 57. ALIGNMENT. 63 

J(163.4 + 163.8) = 163.6; A, |-(168.0 + 1G2.G) = 103.1. Tlien 
eh=^ 0.5. The elevations of the points on the curve are: 

Sta. 15 + 20, {A) 1G3.4 

'' 16 , 163.-1- (.80 X 0.8) + (.SO' X 0.5) = 163.08 

" 17 , 162.6 + (.80 X 1.2) + (.20' X 0.5) = 163.58 

u 17_|_20, {C) 163.8 

A theoretical inaccuracy in tlie above method lies in the fact 
that eh and all parallel lines are not truly vertical. In the 
above case the variation from the vertical is 0° 07^, while tlie 
effect of this variation on the elevations in this case (as in the 
most extreme cases) is absolutely inappreciable. The grades 
in the figure are necessarily very greatly exaggerated, which 
increases the apparent inaccuracy. 



CHAPTER III, 

EARTHWORK. 

FORM OF EXCAVATIONS AN^D EMBANKMENTS. 

58. Usual form of cross- section in cut or fill. The normal 
form of cross-section in cut is as sliown^^in Fig. 37, in which 
e . . . g represents the natural surface of the ground, no matter 
how irregular; ab represents the position and width of the re- 



e \ 




quired roadbed ; ac and hd represent the ' ' side slopes ' ' which 
begin at a and h and which intersect the natural surface at such 




d 



Fig. 38. 



points {g and d^ as will be determined by the required slope 
angle (^). 



64 



g 00. EARTHWORK. 65 

The normal section in iill is as shown in Fig. 38. The points 
c and d are likewise determined by the intersection of the re- 
quired side slopes with the natural surface. In case the required 
roadbed {ah in Fig. 39) intersects the natural surface, both cut 




Fig. 39. 



and fill are required, and the points c and d are determined as 
before. Is^'ote that ^ and /?' are not necessarily equal. Their 
j)roper values will be discussed later. 

59. Terminal pyramids and wedges. Fig. 40 illustrates the 
general form of cross-sections when there is a transition from 
cut to fill, a . . . g represents the grade line of the road which 
passes from cut to fill at d. sdt represents the surface profile. 
A cross-section taken at the point where either side of the road- 
bed first cuts the surface (the point m in this case) will usually 
be triano-ular if the ground is regular. A similar cross-section 
should be taken at o^ where the other side of the roadbed cuts 
the surface. In general the earthwork of cut and fill terminates 
in two pyramids. In Fig. -10 the pyramid vertices are at n 
and h^ and the bases are Ihrn and opq. The roadbed is generally 
wider in cut than in fill, and therefore the section Ihm and the 
altitude In are generally greater than tlie section oj^q and the 
altitude ^A:'. When the line of intersection of the roadbed and 
natural surface {nodkm) becomes perpendicular to the axis of 
the roadbed {ag) the pyramids become wedges whose bases are 
the nearest convenient cross- sections. 

60. Slopes, a. Cuttings. The required slopes for cuttings 
vary from perpendicular cuts, which may be used in hard rock 
which will not disintegrate by exposure, to a slopeof perhaps 



66 



RAILROAD CONSTRUCTION. 



60. 



4 horizontal to 1 vertical in a soft material like quicksand or in 
a clayey soil winch flows easily when saturated. For earthy 
materials a slope of 1 : 1 is the maximum allowable, and even 
this should only be used for firm material not easily affected by 




Fig. 40. 



saturation. A slope of IJ horizontal to 1 vertical is a safer 
slope for average earthwork. It is a frequent blunder that 
slopes in cuts are made too steep, and it results in excessive work 
in clearing out from the ditches the material that slides down, 
at a much higher cost per yard than it would have cost to take 
it out at first, to say nothing of the danger of accidents from 
possible landslides. 

b. Embankments. The slopes of an embankment vary from 
1 : 1 to 1.5 : 1 . A rock fill will stand at 1:1, and if some care 
is taken to form the larger pieces on the outside into a rough 
dry wall, a much steeper slope can be allowed. This method is 
sometimes a necessity in steep side-hill work. Earthwork em- 
bankments generally require a slope of 1 J to 1. If made 
steeper at first, it generally results in the edges giving way, re- 
quiring repairs until the ultimate slope is nearly or quite IJ- : 1. 
The difficulty of incorporating the added material with the old 
embankment and preventing its sliding off frequently makes 
these repairs disproportionately costly. 



§ 62. EARTIIWOIiK. 67 

61. Compound sections. AVlien the cut consists partly of 
earth and partly of rock, a compound cross-section must be 




Fig. 41. 

made. If borings have been made so that tlie contour of the 
rock surface is accurately known, then the true cross-section may 
be determined. The rock and earth should be calculated sepa- 
rately, and this will require an accurate knowledge of where the 
rock "runs out" — a difficult matter when it must be deter- 
mined by boring. During construction the center part of the 
earth cut would be taken out first and the cut widened until a 
sufficient width of rock surface had been exposed so that the 
rock cut would have its proper width and side slopes. Then the 
earth slopes could be cut down at the proper angle. A " berni " 
of about three feet is usually left on the edges of the rock cut as 
a margin of safety against a possible sliding of the earth sloj^es. 
After the work is done, the amount of excavation that has been 
made is readily computable, but accurate preliminary estimates 
are difficult. The area of the cross- section of earth in the figure 
must be determined by a method similar to that developed for 
borrow-pits (see § 89). 

62. Width of roadbed. Owing to the large and often dis- 
proportionate addition to volume of cut or fill caused by the ad- 
dition of even one foot to the width of roadbed, there is a 
natural tendency to reduce the width until embankments become 
unsafe and cuts are too narrow for proper drainage. The cost 
of maintenance of roadbed is so largely dependent on the drain- 
age of the roadbed that there is true economy in making an 



68 



RAILROAD CONSTRUCTION, 



§63.. 



ample allowance for it. The practice of some of the leading 
railroads of the country in this respect is given in the following 
table, in which are also given some data belonging more properlj 
to the subject of superstructure. 



WIDTH OF ROADBED FOR SINGLE AND DOUBLE TRACK-SLOPE RATIOS- 
DISTANCES BETWEEN TRACK CENTERS. 



Road. 



A., T. & Santa Fe. ... 

Chi., Burl. & Quincy 
Chi., Mil. & St. Paul. 
C, C, C. &St. Louis 
Illinois Central. . — 

Erie ... 

Lehigh Valley 

L. S. & Michigan So. 
Louisville & Nashv. . 
Michigan Central. . . 
N. Y. N. H. &H.... 



Norfolk & Western... 

Pennsylvania -j 

Union Pacific 



Single Track. 



Cut. 



j 28' earth 
1 2-i' rock 
14 + (2 X.5) * 
18 + (:i X 6) 
20 + (2 X 4) 
32.5 
20' 81^" 
14 + (2 X3.5) 



13 + (2X4.5) 



21' 2" earth 



16' 



rock 



19' 2" light traffic 
27' 2" heavy " 
14 -i- (2 X 3.5) 



Fill. 



20 



16 
20 to 24 

20 

18 
20' 81^" 

16 



16 



17' 2' 



19' 2" 

19' 2" 

16 



Double Track. 



Cut. 



28 + (2 X 5) 
31 + (2 X 6) 
33 + (2X4) 



33' 81^" 
27 + (2X3.5) 
33 + (8X7.25; 

33 + (2X2.5) 

30 
34' 2" earth 

29' rock 



31' 4"+(2 X 4) 



Fill. 



30 

33 to 37 

33 



33' 81^" 
30 
32 



33 

30 

30' 2' 



31' 4" 



Slope 
Ratios. 



Cut. 



1 : 

Va 
1.5 
1.5 
1.5 
1.5 
1.5 

1 : 
1.5 

1 : 
1.5 
1.5 
1 5 



1.5: 1 
1 : 1 



Fill. 



1.5: 1 



1.5 
1.5 
1.5 
1.5 
1.5 
1.5 
1.5 
1.5 
1.5 
1.5 
1.5 
I 



1.5: 1 
1.5 : 1 



J? c 



.2£E-i 



14' 
13' 
13' 

13' 
13' 
13' 

13' 
12' 
13' 

13' 
12' 2" 



* (2 X 5) signifies two ditches each 5 feet wide: the following cases should be interpreted 
similarly. 

It may be noted from the above table that the average width 
for an eartliworh cut, single track, is about 24.7 feet, with a 
minimum of 19 feet 2 inches. The widths of fills, single track, 
averasre over 18 feet, with numerous minimums of 16 feet. 
The widths for double track may be found by adding the distance- 
between track centers, which is usually 13 feet. 

63. Form of subgrade. The stability of the roadbed depends 
largely on preventing the ballast and subsoil from becoming 
saturated with water. The ballast must be porous so that it 
will not retain water, and the subsoil must be so constructed that it 
will readily drain off the rain-water that soaks through the ballast. 
This is accomplished by giving the subsoil a curved form, convex 



§ 64. EARTHWORK. 69 

upward, or a surface made up of two or three planes, the two 
outer phines liaviiig a slope of about 1 : 21 (sometimes more 
and sometimes less, de23ending on the soil) and the middle plane, 
if three are used, being level. When a circular form is used, 
a crownini^ of 6 inches in a total width of 17 or 18 feet is iren- 
erally used. Occasionally the subgrade is made level, especially 
in rock-cuts, but if the subsoil is previously compressed by 
rolling, as required on the X. Y. C. cV: H. K. E. E., or if the 
subsoil is drained by tile drains laid underneath the ditches, the 
necessity for slopes is not so great. Eock cuts are generally 
required to be excavated to one foot below subgrade and then 
tilled up again to subgrade with the same material, if it is suit- 
able. 

64. Ditches. ' ' The stability of the track depends upon the 
strength and permanence of the roadbed and structures upon 
wliich it rests ; whatever will protect them from damage or pre- 
vent premature decay should be carefully observed. The worst 
enemy is water, and the further it can be kept away from the 
track, or the sooner it can be diverted from it, the better the 
track will be protected. Cold is damaging only by reason of 
the water which it freezes ; therefore the first and most impor- 
tant provision for good track is drainage." (Eules of the Eoad 
Department, Illinois Central E. E.) 

The form of ditch generally prescribed has a flat bottom V2" 
to 24'' wide and with sides having a minimum slo])e, except in 
rock- work, of 1:1, more generally 1.5 : 1 and sometimes 2:1. 
Sometimes the ditches are made Y-shaped, which is objection- 
able unless the slopes are low. The best form is evidently that 
which will cause the greatest flow for a given slope, and this 
will evidently be the form in which the ratio of area to wetted 
perhneter is the largest. The semicircle ful- 
fills this condition better than any other 
form, but the nearly vertical sides would be 
difiicult to maintain. (See Fig. 42.) A ditch, Fig. 42. 

with a flat bottom and such slopes as tlie soil requires, which 
approximates to the circular form will therefore be the best. 




70 RAILROAD COJSSTRUCTION. % Qb. 

When the flow will probably be large and at times rapid it 
will be advisable to pave the ditches with stone, especidly if the 
soil is easily washed away. Six-inch tile drains, placed 2^ under 
the ditches, are prescribed on some roads. (See Fig. 43.) l^o 
better method could be devised to insure a dry subsoil. The 
ditches through cuts should be led off at the end of the cut so 
that the adjacent embankment will not be injured. 

Wherever there is danger that the drainage from the land 
above a cut will drain down into the cut, a ditch should be made 
near the edge of the cut to intercept this drainage, and this 
ditch should be continued, and paved if necessary, to a point 
where the outflow will be harmless. I^eglect of these simple 
and inexpensive precautions frequently causes the soil to be 
loosened on the shoulders of the slopes during the progress of a 
heavy rain, and results in a landslide which will cost more to 
repair than the ditches which would have prevented it for all 
time. 

Ditches should be formed along the bases of embankments ; 
they facilitate the drainage of water from the embankment, and 
may prevent a costly slip and disintegration of the embankment. 

65. Effect of sodding the slopes, etc. Engineers are unani- 
mously in favor of rounding off the shoulders and toes of 
embankments and slopes, sodding the slopes, paving the ditches, 
and providing tile drains for subsurface drainage, all to be put 
in during original construction. (See Fig. 43.) Some of the 
highest grade specifications call for the removal of the top layer 
of vegetable soil from cuts and from under proposed fills to 
some convenient place, from which it may be afterwards spread 
on the slopes, thus facilitating the formation of sod from grass- 
seed. But while engineers favor these measures and their 
economic value may be readily demonstrated, it is generally 
impossible to obtain the authorization of such specifications 
from railroad directors and 2)i*C)moters. The addition to the 
original cost of the roadbed is considerable, but is by no means 
as great as the capitalized value of the extra cost of mainte- 
nance resulting from the usual practice. Fig. 43 is a copy of 



^e5. 



EAHTUWORK. 



71 




PROPOSED SECTION OF ROADBED IN EXCAVATION. 




CUSTOMARY SECTION OF ROADBED ON EMBANKMENT. 

GRAVELn 




PROPOSED SECTION OF ROADBED ON EMBANKMENT. 

GRAVEL, 




Fig. 43.— " VVhittemoke on Railway Excavation and Embankments,' 
Trans. Am. Soc. C. E., Sept. 1894 



72 RAILROAD CONSTRUCTION. § 66. 

designs * presented at a convention of the American Society of 
Civil Engineers by Mr. D. J. Wliittemore, Past President of 
the Society and Chief Engineer of the Chi., Mil. tfe St. Paul 
R.R. The " customary sections " represent what is, with some 
variations of detail, the practice of many railroads. The '' pro- 
posed sections" elicited unanimous approval. They should be 
adopted when not prohibited by financial considerations. 



EAETHWOKK SURVEYS. 

66. Relation of actual volume to the numerical result. It 
should be realized at the outset that the accuracy of the result 
of computations of the volume of any given mass of earthwork 
has but little relation to the accuracy of the mere numerical 
work. The process of obtaining the volume consists of two 
distinct parts. In the first place it is assumed that the volume 
of the earthwork may be represented by a more or less com- 
plicated geometrical form, and then, secondly, the volume of 
such a geometrical form is computed. A desire for simplicity 
(or a frank willingness to accept approximate results) will often 
cause the cross-section men to assume that the volume may be 
represented by a very simple geometrical form which is really 
only a very rough approximation to the true volume. In such 
a case, it is only a waste of time to compute the volume with 
minute numerical accuracy. One of the first lessons to be 
learned is that economy of time and effort requires that the 
accuracy of the numerical work should be kept proportional to 
the accuracy of the cross-sectioning work, and also that the 
accuracy of both should be proportional to the use to be made 
of the results. The subject is discussed further in § 94. 

67. Prismoids. To compute the volume of earthwork, it is 
necessary to assume that it has some geometric form whose vol- 
ume is readily determinable. The general method is to consider 

* Trans. Am. Soc. Civil Eng., Sept. 1894. 



g 08. EAnrilWORK. 73 

the volume as consisting of a series of jprisinoids, which are 
solids having parallel plane ends and bounded by surfaces which 
may be formed by lines moving continuously along the edges of 
the bases. These surfaces may also be considered as the sur- 
faces generated by lines moving along the edges joining the cor- 
responding points of the bases, these edges being the directrices, 
and the lines being always parallel to either base, which is a 
plane director. The surfaces thus developed may or may not 
be planes. The volume of such a prismoid is readily determi- 
nable (as explained in g 70 et seq.), while its definition is so very 
general that it may be applied to very rough ground. The 
" two ])lane ends " are sections perpendicular to the axis of the 
road. The roadbed and side slopes (also plane) form three of 
the side surfaces. The only approximation lies in the degree of 
accuracy with which the plane (or warped) surfaces coincide with 
the actual surface of the ground between these two sections. 
This accuracy will depend (a) on the number of points whicli 
are taken in each cross-section and the accuracy with which the 
lines joining these points coincide with the actual cross-sections ; 
(h) on the skill shown in selecting places for the cross-sections so 
that the warped surfaces shall coincide as nearly as possible with 
the surface of the ground. In fairly smooth country, cross- 
sections every 100 feet, placed at the even stations, are suf- 
ficiently accurate, and such a method simplifies the computations 
greatly ; but in rough country cross-sections must be inter- 
polated as the surface demands. As will be exj^lained later, 
carelessness or lack of judgment in cross-sectioning will introduce 
errors of such magnitude that all refinements in the computations 
are utterly wasted. 

68. Cross-sectioning. The process of cross-sectioning con- 
sists in determining at any place the intersection by a vertical 
plane of the prism of earth lying between the roadbed, the side 
slopes, and the natural surface. The intersection with the road- 
bed and side slopes gives three straight lines. The intersection 
with the natural surface is in general an irregular line. On 
smooth regular ground or when approximate results are accc])t- 



74 



BAILED AD CONSTRUCTION. 



68. 



able this line is assumed to be strais^ht. Accordins: to the irreo^- 
ularitj of the ground and the accuracy desired more and more 
" intermediate points " are taken. 

The distance {d in Fig. 44) of the roadbed below (or above) 
the natural surface at the center is known or determined from 




Fig. 44. 



the profile or by the computed establishment of the grade line. 
The distances out from the center of all ' ' breaks ' ' are determined 
with a tape. To determine the elevations for a cut, set up a 
level at any convenient point so that the line of sight is higher 
than any point of the cross-section, and take a rod reading on 
the center point. This rod reading added to d gives the height 
of the instrument (H. I.) above the roadbed. Subtracting from 
H. I. tlie rod reading at any ^' break " gives the height of that 
point above the roadbed (A,, hi, h^, etc.). This is true for all 
cases in excavation. For fill, the rod reading at center minus 
d equals the II. I., which may be positive or negative. When 
negative, add to the " H. I." the rod readings of the inter- 
mediate points to get their depths below " grade " ; when posi- 
tive, subtract the " H. I." from the rod readings. 

The heights or depths of these intermediate points above or 
below grade need only be taken to the nearest tenth of a foot, 
and the distances out from the center will frequenth^ be sufii- 



69. 



EAHTUWORK. 



75 



ciently exact when taken to tlie nearest foot. Tlie roughness of 
the surface of farming land or woodland generally renders use- 
less any attempt to compute the volume with any greater accu- 
racy than these ligures would im])ly unless the form of the ridges 
and hollows is especially well delined. The position of the slope- 
stake points is considered in the next section. Additional dis- 
cussion regarding cross-sectioning is found in § 82. 

69. Position of slope-stakes. The slope-stakes are set at the 
intersection of the required side slopes with the natural surface, 
which depends on the center cut or till ((/). The distance of 



1>^ 







y^- >- 


_ ; rp >! 


1 


SI 


y : 

1 . 


1 





Fig. 45. 

the slope-stake from the ceiiter for the lower side is ,r = \lj 
+ s{d -|- y) ; for the up-hill side it is x = JZ* + s{d — y'). 
s is the " slope ratio " for the side slopes, the ratio of horizontal 
to vertical. In the above equation both x and y are unknown. 
Therefore some position must be found by trial which will sat- 
isfy the equation. As a preliminary, the value of x for the 
point a =z ^h -[- sd, which is the value of x for level cross- 
sections. In the case of fills on sloping ground the value of x 
on the doiim-Mll side is greater than this; on the up-liill side it 
is less. The difference in distance is s times the difference of 
elevation. Take a numerical case corresponding with Fig. 45. 
The rod reading on <? is 2.9 ; fZ = -l.^ ; therefore the telescope is 
4.2 — 2.9 = 1.3 leloio grade. ^ = 1.5 : 1, Z* = K). Hence for 
the point a (or for level ground) x — \ Y^ lG-t-1.5x4.2 = 
14.3. At a distance out of 14.3 the ground is seen to be about 3 
feet lower, which will not only require 1.5x3 = 45 more, but 



70. 



'^^ RAILROAD CONSTRUCTION. 

enough additional distance so that the added distance shall be 
1.5 times the additional drop. As a first trial the rod may be 
held at 24 feet out and a reading of, say, 8.3 is obtained. 8 3 
+ 1.3 = 9.6, the. depth of the point below grade. The point 
on the slope line {n) which has this depth below grade is at a 
distance from the center x = 8 + 1.5 X 9.6 = 22 A. The 
point on the surface {s) having that depth is 24 feet out. There- 
fore the true point (m) is nearer the center. A second trial at 
20.5 feet out gives a rod reading of, say, 7.1 or a depth of 8.4 
below grade. This corresponds to a distance out of 20.6. Since 
the natural soil (especially in farming lands or woods) is generally 
so rough that a difference of elevation of a tenth or so may be 
readily found by slightly varying the location of the rod (even 
though the distance from the center is the same), it is useless' to 
attempt too much refinement, and so in a case like the above the 
combination of 8.4 below grade and 20.6 out from center may 
be taken to indicate the proper position of the slope-stake. This 
is usually indicated in the form of a fraction, the distance out 
being the denominator and the height above (or below) grade 
being the numerator; the fact of cut or Jill may be indicated by 
O or F. Ordinarily a second trial will be sufiicient to determine 
with sufiicient accuracy the true position of the slope-stake. 
Experienced men will frequently estimate the required distance 
out to within a few tenths at the first trial. The left-hand pages 
of the note-book should have the station number, surface eleva- 
tion, grade elevation, center cut or fill, and rate of grade. The 
right-hand pages should be divided in the center and show the 
distances out and heights above grade of all points, as is illustrated 
in § 84. The notes should read up .the page, so that when look- 
ing ahead along tlie hue the figures are in their proper relative 
position. The "fractions'' farthest from the center line repre- 
sent the slope-stake points. 

COMPUTATION OF VOLUME. 

70. Prismoidal formula. Let Fig. 46 represent a triangular 
prismoid. The two triangles forming the ends lie in parallel 



§ 70. 



EARTHWORK. 



77 



planes, but since the angles of one triangle are not equal to the 
corresponding angles of the other triangle, at least two of the sur- 
faces must be wa7'j)ed. If a section, parallel to the bases, is 




— -6i— ■ 



Fig. 46. 



made at any point at a distance x from one end, the area of the 
section will evidently be 



«"i 



A, = khX = 4[j. + (^A - ^',)7ji_/'. + (^'> - ^'.)7 J- 

The volume of a section of infinitesimal lengtli will be AJx, and 
the total volume of the prismoia will be * 



7'' X^ 



x'y 



* students unfamiliar with the Integral Calculus may take for granted the 
fundamental formula, th^t fdx = a, that fxdx = l^, and that fx^dx = ix- 
also that in integrating between the limits of I and (zero) the value of the 
integral may be found by simply substituting I for x after integration. 



'^'8 RAILROAD CONSTRUCTION. § 70. 

= ^[4^.^. + iU^ + K) + ^\{K + K) + iKh;] 

= - [J, + 4^,„ + ^J, (45) 



in which ^, , ^^ , and A,,, are the areas respectively of the two 
bases and of the middle section. IS'ote that A^ is not the mean 
of ^land ^., , although it does not necessarily differ very greatly 
from it. 

The above proof is absolntely independent of the values, ab- 
solute or relative, of 5^ , Z>, , h, or h^. For example, h^ may be 
zero and the second base reduces to a line and the prismoid be- 
comes wedge-shaped ; or l^ and h^ may both vanish, the second 
base becoming a point and the prismoid reduces to a pyramid 
Since every prismoid (as defined in § 67) may be reduced to a 
combination of triangular prismoids, wedges, and pyramids, and 
since the formula is true for any one of them individually, it is 
true for all collectively ; therefore it may be stated that ^ 

The volume of a prismoid equals one sixth of the perpendic- 
tdar distance hetween the hases multiplied hy the sum of the 
areas of the two hases plies four times the area of the middle 
section. 

While it is always possible to compute the volume of anv 
prismoid by tlie above method, it becomes an extremely compli- 
cated and tedious operation to compute the true value of tlie 
middle section if the end sections are complicated in form. It 

* The student should note that the derivation of equation (45) does not com- 
plete the proof, but that the statements in the following paragraph are logi- 
cally necessary for a general proof. 



§ 72. EAHTUWOKK. 79 

therefore becomes a simpler operation to compute volumes by ap- 
proximate formult>3 and apply, if necessary, a correction. The 
most common methods are as follows : 

71. Averaging end areas. The volume of the triangular 
prismoid (Fig. -iB), computed by averaging end areas, is 

—[:J^i A, -[-J />,/',]. Subtracting this from the true volume (as 
given in the equation above, Eq. (45) ), we obtain tlie correction 



Y2^iJ\-h:){K-h,)-]. .... (46) 



Thi« shows that if either the A's or 5's are equal, the correc- 
tion vanishes ; it also shows that if the bases are roughly similar 
and h varies roughly with h (which usioally occurs, as will be 
seen later), the correction will be negative^ which means that the 
method of averaging end areas usually gives too large results. 

72. Middle areas. Sometimes the middle area is computed 
and the volume is assumed to be equal to the length times the 

middle area. This will equal - X ' ' X - ' T" ' . Subtract- 

ing this from the true volume, we obtain the correction 



24(^1 - ^.)(/^ - /O (^7) 



As l)efore, the form of the correction shows that if either 
the A's or ^'s are equal, the correction vanishes; also under the 
usual conditions, as before, the correction is positive and only 
one-half as large as by averaging end areas. Ordinarily the 
labor involved in the above method is no less than that of 
applying the exact prismoidal formula. 

73. Two-level ground. When aj)j)roxir)iate computations of 
earthwork are sufficiently exact the field-work may be materi- 
ally reduced by observing simply the center cut (or fill) and the 



80 



RAILROAD CONSTRUCTION. 



§73. 



natural slope «, measured with a clinometer. The area of such 
a section (see Fig. 48) equals 



N % 



t£^ 



-J 



_-_(-r^ 



^\,^ 


^ 


! 

1 


^-^^^^^^^7 






\d' \/ \ / 


. ^^k^^^;==^^^^^^^ 


r 


Fig. 47. 


Fig. 48. 


i{a-\-d){Xi-\-x,)- 


ab 

2* 





But 

from which 

Similarly, 

Substituting, 



soi tan /3 = a -{- d ~\- Xj tan or, 

__ <^ + ^ 
' tan /J — tan a 

a -\- d 



Xj~, — 



Area = {a -\- d) 



tan /? -[- tan a 
tan /? 



^5 



tan^ j3 — tan^ o' 2 * * 



. (48) 



The values a^ tan /?, tan" /? are constant for all sections, so 
that it requires but little work to find the area of any section. 
As this method of cross-sectioning implies considerable approxi- 
mation, it is generally a useless refinement to attempt to com- 
pute the volume with any greater accuracy than that obtained 
by averaging end areas. It may be noted that it may be easily 
proved that the correction to be applied is of the same form as 
that found in § 71 and equals 

^[(«/+ <) - W + <')][(^"+ «) - {d'+ a)], 



§ 74. EARTHWORK. 81 

which reduces to 

^ . M T/ , 7 N t^D /^ / , 7//V tan ^ Tr „/ „-. I 

b|L tun- /:^— tan* a tan'/^— tau'a J^ i 

When cZ" = d' the correction vanislies. This shows that 
when the center heights are equal there is no correction — 
regardless of the slope. If the slope is uniform throughout, 
the form of the correction is simplified and is invariably nega- 
tive. Under the usual conditions the correction is negative^ 
i.e., the method generally gives too large results. 

74. Level sections. AVhen the country is very level or when 
only approximate preliminary results are required, it is some- 
times assumed that the cross-sections are level. The method of 
level sections is capable of easy and rapid convputation. The 
area may be written as 

{a + dp-^- (50) 



/7777mm77m777mm/J/777777777mm7777777g77mm7m 




Fig. 49. 



1 

This also follows from Eq. (48) when « = and tan /? = -. 



5 here represents the " slope ratio," 2. 6., the ratio of the hori- 
zontal projection of the slope to the vertical. A table is very 
readily formed giving the area in square feet of a section of 
given depth and for any given width of roadbed and ratio of 
side-slopes. The area may also be readily determined (as illus- 
trated in the following example) without the use of such a 
table; a table of squares will facilitate the work. Assuming 



S2 RAILROAD CONSTRUCTION. § 75, 

t-he cross-sections at equal distances ^(= Z) apart, tlie total ap- 
proximate volume for any distance will be 

^[^, + 2(^ + ^,+ ...^,^_j + .4j. . . (51) 

The prismoidal correction may be directly derived from 
Eq. (46) as :^\2{a + d')s - 2{a + d")s][{a + d") -{a-\- d')], 
which reduces to 

-^-^{d'-d-'r or -^l{d'-d")\ . (52) 



This may also be derived from Eq. (49), since a = 
tan a = 0^ and tan ^ = 2a ~ h. This correction is always 
negative, showing that the method of averaging end areas 
when the sections are level, always gives too large results. The 
prismoidal correction for any one prismoid is therefore a con- 
stant times the square of a difference. The squares are always 
positive whether the differences are positive or negative. The 
correction therefore becomes 

-I2^a^^'^'~'^"y- • • • . . (53) 



75. Numerical example : level sections. Given the following 
center heights for the same number of consecutive stations 100 
feet apart; width of roadbed 18 feet; slope IJ to 1. 

The products in the fifth column may be obtained very 
readily and with sufficient accuracy by the use of the slide-rule 
described in § 79. The products should be considered as 

{a -f d){a -f 6?) -f- -. In this problem s = 1^,- = .6667, 

^ s 

To apply the rule to the first case above, place 6667 on scale jB 
over 89 on scale A, then opposite 89 on scale B will be found 



§76 



EAHTUWOPdv. 



83 



118.8 on scale A. The position of the decimal point will 
be evident from an approximate mental solution of the prob- 
lem. 



1 

sta. 


Center 
HeiKlit. 


a + d 


(a + dV 


(a + d)2s 


17 
18 
19 
20 
21 
22 


2.9 
4.7 
6.8 
11.7 
4.2 
1.6 


8.9 
10.7 
12.8 
17.7 
10.2 

7.6 


79.21 
114.49 
163.84 
313.29 
104.04 

57.76 


118.81 
171.741 
245.76 . 
469.93 f 
156.06 J 
86.64 



Areas. 



X2=^ 



118.81 
r 343.48 
491.52 
939.86 
1^312.12 
86.64 



d' ~ d" 

' 1 


(d' ~ d")" 


1.8 


3.24 


2.1 


4.41 


i 4.9 


24.01 


i 7.5 


56.25 


2.6 


6.76 



2 



6_XJ8 
2 



= 54 
1752.43x100 



2292.43 
10 X 54 = 540 

1752.43 



94.67 



2X2 
Corr. = — 



= 3245 cub. yards = approx. vol. 
100 X 18 



X 94.67 



= — 91 cub. yds. 



12X6X27 
3245 — 91 = 3154 cub. yds. = exact volume. 

The above demonstration of the correction to be applied to 
the approximate volume, found by averaging end areas, is intro- 
duced mainly to give an idea of the amount of that correction. 
Absolutely level sections are practically unknown, and the error 
involved in assuming any given sections as truly level will 
ordinarily be greater than the computed correction. If greater 
accuracy is required, more points should be obtained in the 
cross-sectioning, which will generally show that tlie sections 
are not truly level. 

76. Equivalent sections. When sections are very irregular 
the following method may be used, especially if great accuracy 
is not required. The sections are plotted to scale and then a 
uniform slope line is obtained by stretching a thread so that the 
undulations are averaged and an equivalent section is ol)tained. 
The center depth (d) and the slope angle {oi) of tliis line can 
be obtained from the drawing, but it is more convenient to 
measure the distances {xi and x^ from the center. The area 



84 RAILROAD CONSTRUCTION. % 76 

may then be obtained, independent of the center depth as 
follows : Let s — the slope ratio of the side slopes = cot fi =. — . 
(See Fig. 48.) Then the 

. JL /mS; "T~ "^r I / \ /■ T I I OjO 






(64) 



The true volume, according to the prismoidal formula, of a 
length of the road measured in this way will be 

I roo/xj ah . . foe/ + x," x^ + xj!' 1 ah\ . xl'xj!' ab-^ 
6 



'I ^r 



ah (xl -\- xl' x^ + V 1 ^^^\ , ocl'x^' 



If computed by averaging end areas, the approximate volume 
will be 



(j Xl Xy OjO Xl Xj, O'O 



Subtracting this result from the true volume, we obtain as the 
correction 

Correction = w-(.^/' — ocl){xJ — Xr'). . . (55) 



This shows that if the side distances to either the rip:ht or 
left are equal at adjacent stations the correction is zero, and 
also that if the difference is small the correction is also small 
and very probably within the limit of accuracy obtainable by 
that method of cross-sectioning. In fact, as has already been 
shown in the latter part of § 75, it will usually be a useless 
refinement to compute the prismoidal correction when the 
method of cross-sectioning is as rough and approximate as this 
method generally is. 



77. 



EARTHWORK. 



S^ 



77. Equivalent level sections. These sloping "two-level" 
sections are sometimes transformed into " level sections of equal 
area," a»d tli^ volume computed by the method of level sections 
(§ 74). But the true volume of a prismoid with sloping ends does 
not agree with tliat of a prismoid with equivalent bases and level 
ends except under special conditions, and wdien this method is 
used a correction nmst be applied if accuracy is desired, although, 
as intimated before, the assumption that the sections have uni- 
form slopes will frequently introduce greater inaccuracies than 
that of this method of computation. The following demonstra- 
tion is therefore given to show the scope and limitations of the 
errors involved in this much used method. 

In Fig. 50, let d^ be the center height which gives an 




Fig. 50. 



equivalent level section. The area will equal {a + 6?,)V — — , 

A 
QC X (to 1) 

which must equal the area given in § 76, -^ -r, 5 = .-r-. 

s "A '2(1 



.-. (a-\- d,ys = 



vLitiU' 



I'^r 



or a -\- d^ = 



Vxix^ 



(.56) 



To obtain d^ directly from notes, given in terras of d and n', 



86 RAILROAD CONSTRUCTION. §77. 

we may substitute the values of x^ and x^ given in § 73, which 
gives 

tan ^ _ a-\- d 

' ^ ' '^ r tan /^ — tan a \\ -^ s tan^ «: 

The true volume of the equivalent section may be repre- 
sented by 

![(« + ^^.')' + <'-^+ "-^) + (« + '^/o^; . 

From this there should be subtracted the volume of the 
^' grade prism " under the roadbed to obtain the volume of the 
cut that would be actually excavated, but in the following com- 
parison, as well as in other similar comparisons elsewhere made, 
the volume of the grade prism invariably cancels out, and so for 
the sake of simplicity it will be disregarded. This expression 
for volume may be transposed to 



f'^ f — 1 






The true volume of the prismoid with sloj^ing ends is (see 
76^ 



[^+<t^4^)(^^±^)i)+'^¥]- 



6 



The difference of the two volumes 



i,^,^. 



(rpf^ f\rY,''rp '\rY,fry, ''yrpf'ry, f' _rp'rv, ' _(}\/r,,'ry> 'ry,''ry, '' rp " rv. "\ 

OS 

= ^{ V^/^'- Vx/'x/y. (58) 

This shows that ^'equivalent level sections" do not in 
general give the true volume, there being an exception when 



^ 78. 



EAUTUWOliK. 



87 



Xi^J' = x['Xr . This condition is fiillilled when the slo])e is 
uniforu], i.e., when a' = tx" . When this is nearly so the error 
is evidently not large. On the other hand, if the slopes are in- 
clined in opposite directions the error may be very considerable, 
particularly if the angles of slope are also large. 

78. Three-level sections. The next method of cross-section- 
\\\^ in the order of complexity, and therefore in the order of 




wmiiijnmih 



Fig. 51. 



accuracy, is the method of three-level sections. The area of the 

section is i{a + d){Wr + w^) — -— , which may be written 

ah . 
\{(i -\- d)w — —^ ^ m which vj — \0r-\- v\. If the volume is 

computed by averaging end areas, it will equal 

I 

-\(a^d')io' -al-\-{a + cl'yD'' - ab\ . . (59) 



If we divide by 27 to reduce to cubic yards, we have, when 
I = 100, 

Yol (,...,,) = U{a + djw' - ^ah + ^a + d'^v" - ^al. 
For the next section 



Yol. 



(//*•• ///) — Tl 



^(a + d'yo''- l\ah + f «(« + d'")io'" - ^ah. 



S8 RAILROAD CONSTRUCTWI^. § 78.. 

For a partial station length compute as usual and multiply 

- , leno^th in feet ^, . . , , 

result by — ^^.^ . ilie prismoidal correction mav be 

^ 100 '^ 

obtained by applying Eq. (46) to each side in turn. For the left 

side we have 

r^[(« + d') — {a-\- d'^)']{w{^ — Wi)^ which equals 

^id' - d"){iv{' - io{). 
For the right side we have, similarly, 

l-{d' - d"){wj' - <). 
The total correction therefore equals 

= i^{d' - d")iw" - to'). 

Reduced to cubic yards, and with I = 100, 

Pris. Corr. = ||-(^' - d'yw''-w'). . . (60) 

When this result is compared with that given in Eq. (55) 
there is an apparent inconsistency. If two-level ground is con- 
sidered as but a special case of three-level ground, it would seem 
as if the same laws should apply. If, in Eq. (55), x/ = Xr'\ 
and x/^ is different from x/, the equation reduces to zero ; but 
in this case d' would also be different from d" ; and since x/ -f- 
x/ would = io\ and xC + x^' = w" in Eq. (60), V3" — %o' would 
not equal zero and the correction would be some finite quantity 
and not zero. The explanation lies in the difference in the form 
and volume of the prismoids, according to the method of the 



§78. 



EARTHWORK. 



89 



formation of tlie warped surfaces. If the surface is supposed to 
be generated bj the locus of a Hue moving parallel to the ends 
as plane directors and along two straight lines lying in the side- 
slopes, then ,T;""^- will ecpial ^(a?/ + a?/'), and it',.""'^- will ecpial 
J(x/ -|- a?/'), but the protile of the center line will not be 
straight and t^"""*- will not equal \{(l' + d"). On the other 
hand, if the surfaces be generated by tioo lines moving parallel 
to the ends as plane directors and along a straight center Hne 
and straight side lines lying in the slopes, a warped surface will 
be generated each side of the center line, which will have uni- 
form slopes on each side of the center at the two ends and no- 
where else. This shows that when the upper surface of earth- 
work is warped (as it generally is), two-level ground should not 
be considered as a special case of three-level ground. This dis- 
cussion, however, is only valuable to explain an apparent incon- 
sistency and error. The method of two-level ground should 
only be nsed when such refinements as are here discussed are of 
no importance as affecting the accuracy. 

The following example is given to illustrate the method of 
three-level sections. 



S, 
m 

17 
18 
+40 
19 
20 


•J. 

C 

O 






a + d 


w 


Yards. 


d' - d" 

-5.5 
-2.6 
+4.3 

+2.7 


w"—w' 

+11.7 
+ 8.7 
-13.4 
-15.1 


at 

^6 

-20 
- 3 
-11 
-13 


14.7 
18.6 
23.1 
17.9 
8.4 


^(■^rv 


.* 

> c 

5 3 

+4 
+4 
+5 
+3 




3A' 


8.1F 

10.7^ 

6.4F 

Z.IF 


10.6F 


0.8F 


7.3 
12.8 
15.4 
11.1 

8.4 


31.1 
42.8 
51.5 
38.1 
23.0 


210 
507 
734 
392 
179 


595 
448 
602 
449 


+1 
+3 
+6 
+2 
+1 


2-,'. 9 
1.5.SF 


8.2 
3.4F 


30.7 

•20.2F 
37.3 

14.0F 


12.1 

4.8F 
14.2 

2AF 


28.0 
5.8 F 


10.1 
0.2F 


15.7 


7.3 



Roadbed, 14' wide in fill. Approx. Vol, =2094 
Slope IJ^ to 1. Pris. corr. = 47 



■47 



^ = iT~ = ~5~ = 4. ( ; 



25^ 
27 



ah = 61. 



True Vol. =2047 (disregarding curv. corr).* 

* For the Derivation of the curvation correction, see § 93. 



+16 



90 RAILROAD CONSTRUCTION. § 79. 

In the first column of yards 

210 = ff(« + d)iv = f 5 X 7.3 X 31.1 ; 
507, 734, etc., are found similarly ; 
595 = 210 - 61 + 507 - 61; 

448 = yVo(507 - 61 + 734 - 61); 
602 = -j-Vo('^34 - 61 + 392 - 61) ; 

449 = 392 - 61 + 179 - 61. 
For the prismoidal correction, 

- 20 = l{{cr - fr'){io" - w') = |f(2.6 - 8.1)(42.8 - 31.1) 

= fK-5-5)(+ii.7). 

For the next Hne, - 3 = -/^o_[|5(_ 2.8)(+ 8.7)], and 
similarly for the rest. The "7^" in the columns of center 
heights, as well as in the columns of " right " and '^ left," are 
inserted to indicate Jill for all those pooints. Cut would be 
indicated by " (7." 

79. Computation of products. The quantities ~{a-\-d)w 

25 
and —ah represent in each case the product of two variable 

terms and a constant. These products are sometimes obtained 

from tables which are calculated for all ordinary ranges of the 

variable terms as arguments. A similar table computed for 

25 

— (d^ — d"){io" —- lo') wiir assist similarly in computing the 

ol 

prismoidal correction. Prof. Charles L. Crandall, of Cornell 
University, is believed to be the first to prepare such a set of 
tables, which were first published in 1886 in "Tables for the 
Computation of Railway and Other Earthwork." Another 



§ 79. EARTHWORK. 91 

easy method of obtaining tliese products is by the nse of a slido 
rule. A slide-rule has been designed by the autlior to accom- 
pany this volume. It is designed particularly for this special 
work, although it may be utilized for many other purposes for 
which slide-rules are valuable. To illustrate its use, suppose 
{ic + d) ^ 28.2, and 2^ = 62.4; then 



25, , -,, 28.2 X 62.4 

~—{a -f- a)iv = . 

27^ ^ 1.08 



Set 108 (which, being a constant of frequent use, is specially 
marked) on the sliding scale {B) opposite 282 on the other scale 
(A), and then opposite 624 on scale J] will be found 1629 on 
scale A, the 162 being read directly and the 9 read by estima- 
tion. Although strict rules may be followed for pointing off 
the final result, it only requires a very simple mental calculation 
to know that the result must be 1629 rather than 162.9 or 
16290. For products less than 1000 cubic yards the result 
may be read directly from the scale; for products between 1000 
and 5000 the result may be read directly to the nearest 10 
yards, and the tenths of a division estimated. Between 5000 and 
10,000 yards the result may be read directly to the nearest 20 
yards, and tlie fraction estimated; but prisms of such volume 
will never be found as simple triangular prisms — at least, an as- 
sumption that any mass of ground was as regular as this would 
probably involve more error than would occur from faulty esti- 
mation of fractional parts. Facilities for reading as high as 
10,000 cubic yards would not have been put on the scale ex- 
cept for tlie necessity of finding such products as |7(^^^-1 X 9.5), 
for example. This product would be read off from the same 
part of the rule as f |(91 X 95). In the first case the ])roduct 
(80.0) could be read directly to the nearest .2 of a cubic yard, 
which is unnecessarily accurate. In the other case, the jirod- 
uct (8004) could only be obtained by estimating -^^^ of a division. 
The computation for the prismoidal correction may be made 



92 MAILROAD CONSTBUCriON, § 80. 

similarly except that the divisor is 3.21 instead of 1.08. For 
example, |f(o.5 X 11. T) = ^liAliiJ. get the 324 on scale 

B (also specially marked like 108) opjDOsite 55 on scale A^ and 
proceed as before. 

80. Five-level sections. Sometimes the elevations over each 
edge of the roadbed are observed when cross-sectioning. These 
are distinctively termed " five level sections." If the center, 
the slope-stakes, and one intermediate point on each side [not 
necessarily over the edge of the roadbed) are observed, it i-s 
termed an " irregular section." The field-work of cross-section- 
ing five-level sections is no less than for irregular sections with 
one intermediate point; the computations, although capable of 
peculiar treatment on account of the location of the intermediate 
point, are no easier, and in some respects more laborious; the 
cross-sections obtained will not in general represent the actual 
cross-sections as truly as when there is perfect freedom in locat- 
ing the intermediate point ; as it is generally inadvisable or un- 
necessary to employ five-level sections throughout the length of 
a road, the change from one method to another adds a possible 
element of inaccuracy and loses the advaniage of uniformity of 
method, particularly in the notes and form of computations. 
On these accounts the method will not be further developed, 
except to note that this case, as well as any other, may be 
solved by dividing the whole prismoid into triangular prismoids, 
computing the volume by averaging end areas, and computing 
the prismoidal correction by adding the comjDuted corrections 
for each elementary triangular prismoid. 

81. Irregular sections. In cross-sectioning irregular sec- 
tions, the distance from the center and the elevation above 
''grade" of every "break" in the cross-section must be 
observed. The area of the irregular section may be obtained 
by computing the area of the trapezoids ^fi've^ in Fig. 44) and 
subtracting the two external triangles. For Fig. 44 the area 
would be 



§81. 

hi + h 



EARTHWORK. 



93 



{^i - yi) + 



h + d d +jr , jr + ^' 



-y/+ 



^r + 



\yr — 2r) 



+ -v+i,, _,,_».(,_ I) -|.(,_ I). 




^rh ^. 



Fig. 44. 

Expanding this and collecting terms, of which many will 
cancel, we obtain 

Area = - ^Xiki + yi(d - hi) + x^ + yAjr - K) 

Jr2M-h)+\(hi + K)\. . . (61) 

An examination of this formula will show a perfect regu- 
larity in its formation which will enable one to write out a 
similar formula for any section, no matter how irregular or how 
many points there are, without any of the preliminary work. 
The formula may be expressed in words as follows : 

Area equals one-half the sum of products obtained as folloics : 

the distance to each slope- stake times the height above grade 
of the point next inside the slope-stake y 

the distance to each intermediate point in turn times the height 
of the point just inside minus the height of the point just outside / 

finally^ one-half the width of the roadbed times the sum of 
the slope-stake heights. 



94 



BAILROAD CONSTRUCTION. 



82. 



If one of the sides is perfectly regular from center to slope- 
stake, it is easy to show that the rule holds literally good. 
The ' ' point next inside the slope-stake ' ' in this case is the 
center; the intermediate terms for that side vanish. The last 
term must always be used. The rule holds good for three -level 
sections, in which case there are three terms, which may be 
reduced to two. Since these two terms are both variable quan- 
tities for each cross-section, the special method, given in § 78, 

in which one term (-^J is a constant for all sections, is pref- 
erable. In the general method, each intermediate "break" 
adds another term. 

82. Volume of an irregular prismoid. If there is a break at 
one cross-section which is not represented at the next, the ridge 
(or hollow) implied by that break is supposed to ' ' vanish ' ' at 
the next section. In fact, the volume will not be correctly 




Fig. 52. 



represented unless a cross- section is taken at the point where 
the ridge or hollow "vanishes" or "runs out." To obtain 
the true prism oidal correction it is necessary to observe on the 
ground the place where a break in an adjacent section, which 
is not represented in the section being taken, runs out. For 
example, in Fig. 52, the break on the left of section A" . at a 



g S3. EARTHWORK. 95 

distance of y/'from the center, is observed to run out in section 
A' at a distance of yi from the center. The vohime of the 
prismoid, computed by the prismoidal fornmkx as in § 70, will 
involve the midsection, to obtain the dimension of which will 
require a hiborious computation. A simpler process is to compute 
the volume by averaging end areas as in § 81 and apply a 
prismoidal correction. To do this write out an expression for 
each end area similar to that given in Eq. 61. The sum of 

these areas times -r gives the approximate volume. As before, 

- . - . - - 1 . -1 1 11 length in feet 
lor partial station lengths, multiply the result by — ^7;?; • 

There will be no constant subtractive term, f f«^, as in § 78. 
The true prismoidal correction may be computed, as in § 83, or 
the following approximate method may be used : Consider tlie 
irregular section to be three-level ground for the purpose of 
computing the correction only. This has the advantage of less 
labor in computation than the use of the true prismoidal correc- 
tion, and although the error involved may be considerable in 
individual sections, the error is as likely to be positive as nega- 
tive, and in the long run the error will not be large and generally 
will be much less than would result by the neglect of any 
prismoidal correction. 

83. True prismoidal correction for irregular prismoids. As 
intimated in § 82, each cross-section should be assumed to have 
the same number of sides as the adjacent cross-section when 
computing the prismoidal correction. This being done, it per- 
mits the division of the whole prismoid into elementary triangu- 
lar prismoids, the dimensions of the bases of which being given 
in each case by a vertical distance above grade line and l)y the 
horizontal distance between two adjacent breaks. Tlie summa- 
tion of the prismoidal corrections for each of the elementary 
triangular prismoids will give the true prismoidal correction. 
Assuming for an example the cross-section of Fig. 44, witli a 
cross-section of the same number of sides, and witli dimensions 



96 RAILROAD CONSTRUCTION. §83. 

similarly indicated, for the other end, the prismoidal correction 
becomes (see Eq. 46) 



l4 (^^/- V 



)[{xi" - yi") - {xi - 2/01 + {ki' - ki")[{xi" - yi") - (xi' - yn] 
+ (ki' - ki"){yr - y{) + {d' - cV){y{' - yi') + {d! - d"){Zr" - Zr') 

+ {>' - >'0(2r" - Zr!) + OV -jr")\_{yr" - Zr") - (^r' - s/)l 

+ {kr' - kr")[{yr" - Zr") - {yr' - 2r')l 

+ (A;/-V')[(^V''-2/r'')--(3V'-2/r')l + (Ar'-/^r'')[(2;r''-yr'')-CV-yr')] 

Expanding this and collecting terms, of which many will 
cancel, we obtain 

Pris. Corr. = -^^xf -x{\k{ - h") + {yi" - yi')[{d' - hi') - {d" - Jh")] 

+ iXr" - Xr'){kr' - kr") + {yr" - yr')[{jr' - hr) - {jr" - h/')] 
+ (Zr" -er')[{d' -kr')-{d" -kr")]] (62) 

By comparing this equation with Eq. 61 a remarkable 
coincidence in the law of formation may be seen, which enables 
this formula to be written by mere inspection and to be applied 
numerically with a minim'um of labor from the computations for 
end areas, as will be shown (§ 84) by a numerical example. 
For each term in Eq. 61, as, for example, yXjr — /^r)? there is 
a correction term in Eq. 62 of the form 

ivr" - 2//)[ov' - V) - ijr" - wn ■ 

Each one of these terms (^z/', 1/r\ (J/ — h/), and (J/' — /?/') ) 
has been previously used in finding the end areas and has its 
place in the computation sheet. The summation of the products 
of these differences times a constant gives the total true pris- 
moidal correction in cubic yards for the whole prismoid considered. 
The constant is the same as that computed in § 78, i.e., ||^. 



§84. 



EAHTHWORK. 



97 



84. Numerical example ; irregular sections ; volume^ with true 
prismoidal correction. 



Sta. 



19 

18 

17 

+ 43 

16 



\ cut 

Ceiiter-s <>r 

fill. 



0.6c 



2.3c 



,6c 



10.3c 



6.8c 



Left. 



3.6c 
1^174 

4 .2c 
1573 

8.2c 
3l73 

]2.2c 

27.3 

8.9c 
3274 



^8.2/ \6.0/ 



6^c 

sTi 

l(h2c 
TtTI" 

|12.3c\ 
\2270/ 



3^2 c 

y.2 

8.0c 
6.1 

12.6c 
8.2 

7.6c 



12.0 



Right. 



0.1c 
472 



/1^9cV 
V3.6/ 

|5^c| 

\8.0/ 



6.2c 
775" 

3.2c 



0.4c 
9:6" 

1.2c 
U) 8 

4.2c 
1573 

8.4c 
2176 

2.6c 
12.9 



Koadbed IS feet wide in cut; slope 1|^ to 1. 

The figures in the bracket \-^^-7v j mean that it was noted in 

the field that the break, indicated at Sta. 17 as being 17.4 to 
tlie left, ran out at Sta. 10 + 42 at 22.0 to the left. By inter- 
polation between 8.2 and 27.3 the height of this ])oint is 
<!omjputed as 12.3. The quantities in the other l)rackets are 
obtained similarly. These quantities are only used when the 
computation of the true prismoidal correction is desired. They 
are not needed in computing the volume by averaging end 
areas, nor are they used at all if the prismoidal correction is to 
he obtained by assuming {fov this jnirpose) the ground to be 
three-level ground. 

lu the tabular form on page 98 the figures within the braces 
(- — r- — ) are not used in comjniting the volume, but are only 
used to obtain the differences of widths or heights with which to 
compute the true prismoidal correction. It may be noted, as a 
check, that the volume, computed from these figures in the 
braces, is the same as that computed from the other figures. 



98 RAILROAD CONSTRUCTION. §84. 

VOLUME OF IRREGULAR PRISMOID, WITH TRUE PRISMOIDAL CORRECTION. 













True pris. corr. 


Sta. 


Width. 


Height. 


Vn '"^1 " 










iv" — w' 


h' - h" 


Yards. 




J r22.4 
^ 12.0 


7.6 


158 












- 2.1 


-23 










16 


4.1 ^^ 


3.2 


40 












4.2 


16 












9.0 


11.5 


96 












T r27.3 

^ 8.2 


12.6 


319 




+ 4.9 


-5.0 


-7 




- 2.0 


-15 




-3.8 


-0.1 







i 27.3 


12.3 












+ 42 


L-^22.0 

( 8.2 


0.4 
- 2.1 














21. 6-] j^ 
7.5 ^^ 


6.2 


124 




+ 8.7 


-3 


-8 




1.8 


13 




+ 3.4 


+ 2.4 


+ 3 





9.0 


20.6 


172 


378 






(-5) 


r21.3 
L 17.4 


10.2 


201 




- 6.0 


+ 2.1 


-4 




- 0.2 


- 3 




- 4.6 


+ 0.6 


- 1 




_ 6.1 


- 2.6 


-14 




- 2.1 


+ 0.5 





17 


15.3[j^ 

8.0^^ 


5.8 






- 6.3 


+ 0.4 


- 1 




3.4 






^0.5 


-1.6 







15.3]R 


7 6 


107 












9.0 


12.4 


103 


584 






(-3) 
-6 






rl5.3 


6.8 


95 




- 6.0 


+ 3.4 




L 


8.4 


- 1.0 


,-7 
i 




-9.0 


+ 0.8 


- 2 






u 5.2 


- 4.5 


-22 




-0.9 


+ 1.9 


- 1 


18 


10.8]R 
10.8^ R 
3.6 f" 


2.3 

1.9 


23 




-4.5 


+ 5.3 


-7 




1.1 














9.0 


5.4 


45 


528 






(-16) 




L[14.4 


0.6 


8 












( 14.4 


2.3 






-0.9 


+ 4.5 


-1 




L-^ 8.2 


- 1.8 






-0.2 


+ 0.8 





19 


6.0 


- 1.7 






+ 0.8 


-2.8 


- 1 




4.2 ^^ 


0.1 


1 




-1.2 


+ 1.8 


- 1 




0.2 


1 




+ 0.6 


+ 0.9 







9.0 


4.0 


33 


177 






(-3) 






Approx. vol. = 


: 1667 




- 27 






rrue pris. corr. = 


: - 27 








1 


True volume = 


1640 ci] 


bic yards 





The figures within each brace (or bracket) constitute a group 
wliich must be used in connection with a group which has the 
same number of points, on the same side of the center, in the 
next cross-section, previous or succeeding. In the cohimn of 



§ 86. 



EARTUWORK. 



99 



^'Yards'' under "True pris. corr.,'' we have, for example, 

85. Volume of irregular prismoid, with approximate prismoidal 
correction. It" the prismoidal correction is obtained a])proxi- 
mately, by the method outlined hi § 82, the process will be as 
shown in the tabular form. Kot only is the numerical work 
considerably less than the exact method, but the discrepancy in 
cubic yards is almost insignilicant. 



Sta. 


Widtli. 


Height. 


Yards. 


Cen. 
Height. 


Total 
width. 


d'-d" 


to" — ?«' 


Ai)prox. 
piis. corr. 




22.4 


7.6 


158 




+ 6.8 


35.3 










12.0 


-2.1 


- 23 














16 


12 9 
4.1 


3.2 
4.2 


40 
16 

















9.0 
27.3 


11.5 


96 

319 




+ 10.2 


48.9 








12.6 


-3.4 


+ 13.6 


- 14 




8.2 


-2.0 


- 15 














+ 42 


21.6 
7.5 


6.2 

1.8 


124 
13 
















9.0 
21.3 


20.6 


172 
201 


378 




36.6 


+2.6 




(-6) 


10.2 


+ 7.6 


- 12.3 


- 10 




17.4 


-0.2 


- 3 














17 


6.1 
15 3 


-2.6 

7.6 


- 14 

107 

















9.0 


12.4 


103 
9o 


584 




26.1 


+5.3 




(-6) 


15.3 


6.8 


+ 2.3 


- 10.5 


- 17 




8.4 


-1.0 


- 7 














18 


5.2 

10.8 


-4.5 
2.3 


— 22 
















9.0 


5.4 


45 
8 


528 










(-17) 
- 1 




14.4 


0.6 


+ 0.6 


24.0 


+1.7 


-2.1 


19 


9.6 


0.1 


1 














4.2 


0.2 


1 
















9.0 


4.0 


33 


177 










(-1) 



Approx. volume = 1667 
Approx. pris. corr. = — 30 



- 30 



Corrected volume = 1G37 cubic )'ards 

86. Illustration of value of approximate rules. The accom- 
panying tabulation shows that when the volume of an irregular 
prismoid is computed by averaging end areas and is corrected 
by considering the ground as three-level ground {for the j>ur' 



u tfa 



100 



RAILROAD CONSTRUCTION. 



87. 



poses of the correction only\ the error for tlie different sections 
is sometimes positive and sometimes negative, and in tins case 



Sections. 


6 

s 

3 

O 
> 

0) 
3 
U 

H 


Approx. vol. 
by averaging 
end areas. 


Difference or 
true pris. 
corr. 


Approx. pris. 
corr. on basis 
of three-level 
ground. 


Error. 


Approx. vol., 
computed 
from center 
and side 
heights onhj. 


Error. 


16 16 + 42 

16 + 42. ..17 

17 18 

18 19 


373 

581 

174 


378 
584 
528 
177 

1667 


- 5 

- 3 

- 16 

- 3 


- 6 

- 6 

- 17 

- 1 


- 1 

- 3 

- 1 

+ 2 

- 3 


396 

577 
463 
147 


+ 23 

- 4 

- 49 

- 27 

- 57 




1640 


- 27 


- 30 


15S3 



amounts to only 3 yards in 1640 — less than \ of 1^. If the 
prismoidal correction had been neglected, the error would have 
been 27 yards — nearly li. The approximate results are here 
too large for each section — as is usually the case. If points 
between the center and slope stakes are omitted and the volume 
computed as if the ground were three-level ground, the error is 
quite large in individual sections, but the errors are both posi- 
tive and negative and therefore compensating. 

87. Cross-sectioning irregular sections. The prismoids con- 
sidered have straight lines joining corresponding points in the 
two cross-sections. The center line must be straight between 
tAvo cross-sections. If a ridge or valley is found lying diago- 
nally across the roadbed, a cross-section m^icst be interpolated at 
the lowest (or highest) point of the profile. Therefore a ' ' break ' * 
at any section cannot be said to run out at the other section on 
the opposite side of the center. It must run out on the same 
side of the center or possibly at the center. Yery frequently 
complicated cross-sectioning may be avoided by computing the 
volume, by some special method, of a mound or hollow when 
the ground is comparatively regular except for the irregularity 
referred to. 

88. Side-hill work. When the natural slope cuts the roadbed 
there is a necessity for both cut and fill at the same cross-section. 
When this occurs the cross-sections of both cut and fill are often 
so nearly triangular that they may be considered as such without 



88. 



EARTUWORK, 



101 



great error, and the volumes inaj be computed separately as 
triangular prismoids without adopting tlie more elaborate form 
of computation so necessary for complicated irregular sections. 
AVhen the ground is too irregular for this the best plan is to 
follow the uniform system. In computing the cut, as in Fig. 53, 




Fia. 53. 

the left side would be as usual ; there would be a small center 
cut and an ordinate of zero at a short distance to the right of the 
center. Then, ignoring the fill ^ and applying Eq. 61 strictly, 
we have two terms for the left side, one for the right, and the 
term involving \h^ which will be ^lii in this case, since li^ = 0, 
and the equation becomes 

Area = \\xihi -f- yifl — ^i) + x^d + ^Jfhj']. 
The area for fill may also be computed by a strict application 




Fig. 54. 



of Eq. 61, but for Fig. 54 all distances for the left side are zero 
and tlie elevation for the first point out is zero, d also must be 



102 BAILROAD CONSTRUCTION. § 89. 

considered as zero. Following the rule, § 81, litexallj, the 
equation becomes 

Area(Fin) = ^[x^h + yr{o — K) + z,.{o — h) + iHo + A,-)]? 

which reduces to 

{Note that Xy^ Kj etc., have different significations and 
values in this and in the preceding paragraphs.) The "terminal 
pyramids ' ' illustrated in Fig. 40 are instances of side-hill work 
for very short distances. Since side-hill work always implies 
hoth cut and fill at the same cross-section, whenever either the 
cut or fill disappears and the earthwork becomes wholly cut or 
wholly fill, that point marks the end of the ' ' side-hill work, ' ' 
and a cross-section should be taken at this point. 

89. Borrow-pits. The cross-sections of borrow-pits will vary 
not only on account of the undulations of the surface of the 




'JillHlllllllllllUIIIHNIIIillllllllllWlllWllinillllllllliilllllUIIIWIUIIWIIili' 

Fig. 55. 

ground, but also on the sides, according to whether they are 
made by widening a convenient cut (as illustrated in Fig. 55) 
or simply by digging a pit. The sides should always be prop- 
erly sloped and the cutting made cleanly, so as to avoid un- 
sightly roughness. If the slope ratio on the right-hand side 
(Fig. 55) is ,9, the area of the triangle is ^sm^. The area of the 
section is 2^119 -\-{g-\-h)v-\-{h-\-j)x-^{j-\-li')y-\-{k-\-m)z — smj''\. 
If all the horizontal measurements were referred to one side as 
an origin, a formula similar to Eq. 61 could readily be devel- 
oped, but little or no advantage would be gained on account of 
any simplicity of computation. Since the exact volume of the 
€arth borrowed is frequently necessary, the prismoidal correc- 



§ 90. EARTHWORK. 103 

tiou sliould be computed ; and since such a section as Yi<r. 55 
does not even approximate to a three-level section, thj method 
suggested in § 82 cannot be employed. It will then be neces- 
sary to employ the exact method, § 83, by dividing the volume 
into triangular prismoids and taking the summation of their 
corrections, found according to the general method of § 71. 

90. Correction for curvature. The volume of a solid, iren- 
erated by revolving a plane area about an axis lying in the 
plane but outside of the area, equals the product of the given 
area times the length of the path of the center of gravity of the 
area. If the centers of gravity of all cross-sections lie in the 
center of the road, where the length of the road is measured, 
there is absolutely no necessary correction for curvature. If all 
the cross- sections in any given length were exactly the same 
and therefore had the same eccentricity, the correction for 
curvature would be very readily computed according to the 
above principle. But when both the areas and the eccentrici- 
ties vary from point to point, as is generally the case, a theo- 
retically exact solution is quite complex, both in its derivation 
and application. Suppose, for simplicity, a curved section of 
the road, of uniform cross-sections and w^ith the center of o-rav- 
ity of every cross-section at the same distance e from the center 
line of the road. The length of the path of the center of 
gravity will be to the length of the center line as R ±e\ R. 
Therefore we have True vol. : nominal vol. :: B ± e : R. 

Ti ±e 

.'. True vol. = lA — 73 — for a volume of uniform area and 

eccentricity. For any other area and eccentricity we have, 

R + e' 
similarly, True vol.' = IA'~~—. This shows that the eifect 

of curvature is the same as increasing (or diminishing) tlie area 

by a quantity depending on the area and eccentricity, the 

increased (or diminished) area being found by nuiltiplying the 

.1 ^ 1 . R ± e 

actual area by the ratio — „ — . This being independent of the 

value of Z, it is true for infinitesimal lengths. If the eccen- 



104 RAILROAD CONSTRUCTION. § 91. 

tricity is assumed to vary uniformly between two sections, the 
equivalent area of a cross- section located midway between the 

(^ ± '^) 

two end cross-sections would be A^- ^ -^. There- 

fore the volume of a solid which, when straight, would be 
^(^'4" '^^m -\- ^")^ would then become 

True vol. = ^_^X^ ± e')+4.A,^[R± ^-^^)+A'\B ± e")j. 

Subtracting the nominal volume (the true volume when the 
prismoid is straight), the 

^ r "1 

Correctio7i = ± -^\_{A' + ^A,^)e' + (2^„, + A'y j, (63) 

Another demonstration of the same result is given by Prof. 
C. L. Crandall in his "Tables for the Computation of Rail- 
way and other Earthwork," in which is obtained by calculus 
methods the summation of elementary volumes having variable 
areas with variable eccentricities. The exact application of 
Eq. (63) requires that A^^ be known, which requires laborious 
computations, but no error worth considering is involved if the 
equation is written approximately 

Curv. GOVT. = ^{A'e' + A"e'% . . . (64) 

which is the equation generally used. The approximation con- 
sists in assuming that the difference between A' and A,^^ equals 
the difference between A^^ and A" but with opposite sign. 
The error due to the approximation is always utterly insig- 
niiicant. 

91. Eccentricity of the center of gravity. The determina- 
tion of the true positions of the centers of gravity of a long 
series of irregular cross-sections would be a very laborious 
operation, but fortunately it is generally sufficiently accurate to 



§ 91. EARTHWORK. 105 

consider the cross-sections as three-level ground, or, fur side-hill 
work, to be triangular, /br the purpose of this correction. The 




mmihii\iimh\'mw. 



Fig. 56. 

eccentricity of the cross-section of Fig. 56 (including the grade 
triangle) may be written 

{a^d)xiXi {a-\-d)x^.Xy 
- 2 3 ~ 2 3" _ 1^ Xi' - x; _ 1 

^ ~ (a^d)x, (^5>r ~ 3 xi + X, - 3"^''^" ^^)- (^^) 

2 "^ 2 

The side toward x^. being considered positive in the above 
demonstration, if x^, > Xi^ e would be negative, i.e., the center 
of gravity would be on the left side. Therefore, for three-level 
ground, the correction for curvature (see Eq. 64) may be 
written 

Correction = ^[A'(x/ — cc/) + A"{x/' — x/')]. 
Since the approximate volume of the prismoid is 
^{A + A') = I A' + \a" = r + F", 

in which V^ and V represent the number of cubic yards 
corresponding to the area at each station, we may write 

Corr. in cuh. yds. = ^^ V\xj '- .t/)+ F"(-^/"- ^'")]- (^^<^> 



106 RAILROAD CONSTRUCTION. § 91. 

It should be noted that the value of €, derived in Eq. 65, is 
the eccentricity of the whole area including the triangle under 
the roadbed. The eccentricity of the true area is greater than 
this and equals 

true area -f- i^t^^ 
true area 



e X : -^— = e,. 



The required quantity {A'e of Eq. 64) equals true area X ^n 
which equals {true area -\- ^ah) X e. Since the value of e is very 
simple, while the value of e^ would, in general, be a complex 
quantity, it is easier to use the simple value of Eq. Q^ and add 
^ah to the area. Therefore, in the case of three-level ground 
the subtractive term ^^ah (§ 78) should not be subtracted in 
computing this correction. For irregular ground, when com- 
puted by the method given in §§ 81 and 82, which does not 
involve the grade triangle, a term f |<^5 must be added at every 
station when computing the quantities V and V" for Eq. QQ. 

It should be noted that the factor 1 ~- Sic, which is 
constant for the length of the curve, may be computed with all 
necessary accuracy and without resorting to tables by remember- 
ing that 



deojree of curve 



Since it is useless to attempt the computation of railroad 
earthwork closer than the nearest cubic yard, it will frequently 
be possible to write out all curvature corrections by a simple 
mental process upon a mere inspection of the computation sheet. 
Eq. QQ shows that the correction for each station is of the form 

— P — —. 3i? is generally a large quantity — for a 6° curve 
Sic 

it is 2865. {xi — x^) is generally small. It may frequently be 

seen by inspection that the product Y{Xi — x,) is roughly twice 

or three times 3^, or perhaps less than half of 3^, so that the 

corrective term for that station may be written 2, 3, or cul)ic 

yards, the fraction being disregarded. For much larger absolute 



§92. EARIHWORK. 1()7 

amounts the correction must be computed with a correspondingly 
closer percentage of accuracy. 

The algebraic sign of the curvature correction is best deter- 
mined by noting that the center of gravity of the cross-section is 
on the riglit or left side of tlie center according as x,. is greater 
or less than xi^ and that the correction is positive if the center of 
gravity is on the outside of the cur\'e, and iiegative if on the 
inside. 

It is frequently found that Xi is uniformly greater (or uni- 
formly less) than x,. throughout the length of the curve. Then 
the curvature correction for each station is uniformly positive or 
negative. But in irregular ground the center of gravity is apt 
to be irregularly on the outside or on the inside of the curve, 
and the curvature correction will be correspondingly positive or 
negative. If the curve is to the right, the correction will be 
positive or negative according as {xi — x,.) is positive or negative ; 
if the curve is to the left, the correction ^\'ill be positive or nega- 
tive according as {Xy — xi) is positive or negative. Therefore 
when computing curves to the 7'ight use the form {xi — x,>^ in 
Eqs. 66 and Q^ ; wdien computing curves to the left use the form 
{x^ — x^ in these equations ; the algebraic sign of the correction 
will then be strictly in accordance with the results thus obtained. 

92. Center of gravity of side-hill sections. In computing the 



Fi(.. 57. 

correction for side-hill work the cross section would be treated 
as trianirular unless the error involved would evidentlv be too 



108 



RAILROAD CONSTRUCTION. 



92, 



great to be disregarded. Tlie center of gravity of the triangle 
lies on the line joining the vertex with the middle of the base 
and at ^ of the length of this line from the base. It is therefore 
equal to the distance from the center to the foot of this line plus 
^ of its horizontal projection. Therefore 



e = 



2 ~ 2 \2 + ^' 



+ 



1 r 



Xl 



2 ~2 \2+^' 



•JCy 



Xl 



t-O'V 



- 4 ~ 2 +^ 
h Xl 



3 



3 

Xy 

3 



-^ + {xi-x,)j. 



12+6 



n 



(67) 



Bj the same process as that used in § 91 the correction equation 
may be written 



Corr. in cub. yds. = .IPf^I + (^/ - a-/)) + ^"^2'^^'^'" ~ ""'"^ 



(68) 



It should be noted that since the grade triangle is not used in 
this computation the volume of the grade prism is 7wt involved 
in computing the quantities V and V" . 

The eccentricities of cross-sections in side-hill work are 
never zero, and are frequently quite large. The total volume 
is generally quite small. It follows that the correction for 
curvature is generally a vastly larger proportion of the total 
volume than in ordinary three-level or irregular sections. 

If the triangle is wholly to one side of the center, Eq. 67 
can still be used. For example, to compute the eccentricity of 
the triangle of fill, Fig. 57, denote the two distances to the 
slope-stakes by y,. and — yi (note the minus sign). Applying 

Eq. 67 literally (noting that -^ must here be considered as nega- 
tive in order to make the notation consistent) we obtain 

1 



e = 



3 



- ^+(- 2/^- Vr)]' 



§ 94. EABTIlWOliK. 109 

which reduces to 

in n 
^ = -^[j + yi + f/'j (^>'^) 

As the algebraic signs tend to create confusion in tliese 

forinula3, it is more simple to remember that for a triangle 

lying on hoth sides of the center e is always numerically c(|ual 

lYh n 

to ^ - -{- {xi'^ x^) , and for a triangle entirely on one side, e is 

-j- the numerical su/n of the two dis- 



numerically equal to — 



3 L2 

tances out]. The algebraic sign of e is readily determinable as 
in § 91. 

93. Example of curvature correction. Assume that the fill in 

§ 78 occurred on a 6° curve to the ri(/ht. ^-jj = . The 

quantities 210, 507, etc., represent the quantities V\ V'\ 
etc., since they include in each case the 61 cubic yards due to 
the grade prism. Then 

V(xi — Xr) 210(22.9 - 8.2) _ 3101.7 _ 



3i? - 2865 2865 

The sign is plus since the center of gravity of the cross-sec- 
tion is on the left side of the center and the road curves to the 
right, thus making the true volume larger. For Sta. 18 the 
correction, computed similarly, is -f- 3, and the correction for 
the whole section is 1 -f- 3 = 4. For Sta. 18 -|- 40 the cor- 
rection is computed as 6 yards. Therefore, for the 40 feet, the 
correction is y\%-(3 -[- 6) = 3.6, which is called 4. Computing 
the others similarly we obtain a total correction of -\- 16 cubic 
yards. 

94. Accuracy of earthwork computations. The preceding 
methods give the precise vohc7ne (except where approximations 
are distinctly admitted) of the prismoids which are sujij^osed to 
represent the volume of the earthwork. To appreciate the 
accuracy necessary in cross-sectioning to obtain a given accuracy 



110 JiAIfAlO AD CONSTRUCTION. §94. 

in volume, consider that a fifteen-foot length of the cross-section, 
which is assumed to be straight, really sags 0.1 foot, so that the 
cross-section is in error by a triangle 15 feet wide and 0.1 foot 
high. This sag 0.1 foot high would hardly be detected by the 
eye, but in a length of 100 feet in each direction it would make 
an error of volume of 1.4 cubic yards in each of the two pris- 
moids, assuming that the sections at the other ends were perfect. 
If the cross-sections at both ends of a prismoid were in error by 
this same amount, the volume of that prismoid would be in error 
by 2.8 cubic yards if the errors of area were both plus or both 
minus. If one were plus and one minus, the errors would 
neutralize each other, and it is the compensating character of 
these errors which permits any confidence in the results as 
obtained by the usual methods of cross-sectioning. It demon- 
strates the utter futility of attempting any closer accuracy than 
the nearest cubic yard. It will thus be seen that if an error 
really exists at any cross-section it involves the prismoids on 
hoth sides of the section, even though all the other cross-sections 
are perfect. As a further illustration, suppose that cross-sec- 
tions were taken by the method of slope angle and center depth 
(§ 73), and that a cross-section, assumed as uniform, sags 0.4 
foot in a width of 20 feet. Assume an equal error (of same 
sign) at the other end of a 100-foot section. The error of 
volume for that one prismoid is 38 cubic yards. 

The computations further assume that the warped surface, 
passing through the end sections, coincides with the surface of 
the ground. Suppose that the cross-sectioning had been done 
with mathematical perfection ; and, to assume a simple case, 
suppose a sag of 0.5 foot between the sections, which causes an 
error equal to the volume of a pyramid having a base of 20 feet 
(in each cross-section) times 100 feet (between the cross-sections) 
and a height of 0.5 foot. The volume of this pyramid is 
i(20 X 100) X 0.5 z= 333 cub. ft. = 12 cub. yds. And yet 
this sag or hump of 6 inches would generally be utterly un- 
noticed, or at least disregarded. 

When the ground is very rough and broken it is sometimes 



§96. eartuwohk. Ill 

practically impossible, even with tVeqiient oross-sections, to 
locate warped surfaces which will closely coincides with all the 
sudden irregularities of the ground. In such cases the compu- 
tations are necessarily more or less approximate and dependence 
must be placed on the compensating character of the errors. 

95. Approximate computations from profiles. As a means 
of comparing the relative amounts of earthwork on two or 
more proposed routes which have been surveyed by preliminary 
surveys, it will usually be sufficiently accurate to com})are the 
areas of cutting (assuming that the cut and till are approximately 
balanced) as shown by the several profiles. The errors involved 
may be large in individual cases and for certain small sections, 
but fortunately the errors (in comparing two lines) will be 
largely compensated. The errors are nmch larger on side-hill 
work than when the cross-sections are comparatively level. 
The errors become large when the depth of cut or fill is very 
great. If the lines compared have the same general character 
as to the slope of the cross-sections, the proportion of side-liill 
work, and the average depth of cut or iill, the error involved in 
considering their relative volumes of cutting to be as the relative 
areas of cutting on the profiles (obtained perhaps by a planim- 
eter) will probably be small. If the volume in each case is 
computed by assuming the sections as level^ with a depth espial 
to the center cut, the error involved will depend only on the 
amount of side-hill work and the degree of the slope. if these 
features are about the same on the two lines compared, the error 
involved is still less. 

FORMATION OF EMUANKMKNTS. 

96. Shrinkage of earthwork. The evidence on this subject 
as to the amount of shrinkage is very conflicting, a fact which 
is probably due to the following causes : 

1. The various kinds of earthy material act very differently 
as respects shrinkage. There has been l)ut little uniforniity in 
the classification of earths in the tests and experiments tliat 
have been made. 



112 RAILROAD CONSTRUCTION. § 96. 

2. Yery much depends on tlie method of forming an em- 
bankment (as will be shown later). Different reports have been 
based on different methods — often without mention of the 
method. 

3. An embankment requires considerable time to shrink to 
its final volume, and therefore much depends on the time 
elapsed between construction and the measurement of what is 
supposed to be the settled volume. 

P. J. Fljnn quotes some experiments {Eng. News^ May 1, 
1886) made in India in which pits were dug, having volumes of 
400 to 600 cubic feet. The material, when piled into an em- 
bankment, measured largely in excess of the original measure- 
ment — as is the universal experience. The pits were refilled 
with the same material. As the rains, very heavy in India, 
settled the material in the pits, more was added to keep the pits 
full. Even after the rainy season was over, there was in every 
case material in excess. This would seem to indicate a per- 
manent expansion^ although it is possible that the observations 
were not continued for a sufficient time to determine the final 
settled volume. 

On the contrary, notes made by Mr. Elwood Morris many 
years ago on the behavior of embankments of several thousand 
cubic yards, formed in layers by carts and scrapers, one winter 
intervening between commencement and completion, showed in 
each case a permanent contraction averaging about 10^. 

All authorities agree that rockwork expands permanently 
when formed into an embankment, but the percentages of 
expansion given by different authorities differ even more than 
with earth — varying from 8 to 90^. Of course this very large 
ranore in the coefficient is due to differences in the character of 
the rock. The softer the rock and the closer its similarity to 
earth, the less will be its expansion. On account of the conflict- 
ing statements made, and particularly on account of the influence 
of methods of work, but little confidence can be felt in any 
given coefficient, especially when given to a fraction of a per 



§ 97. . EARTHWORK. 113 

cent, but the consensus of American practice seems to avera"-e 
about as follows : 

Permanent contraction of earth about 10^ 

" expansion of rock 40 to GO^ 

These values for rock should be materially reduced, according 
to judgment, when the rock is soft and liable to disintegrate. 
The hardest rocks, loosely piled, may occasionally give even 
higher results. The following is given by several authors as 
the permanent contraction of several grades of earth : 

Gravel or sand about 8^ 

Clay " lOfo 

Loam " 12^ 

Loose vegetable surface soil '^ 15^ 

It may be noticed from the above table that the harder and 
cleaner the material the less is the contraction. Perfectly clean 
gravel or sand would not probably change volume appreciably. 
The above coefficients of shrinkage and expansion may be used 
to form the following convenient table. 



Material. 


To make 1000 cubic 

yards of embankment 

will require 


1000 cubic yards measured 
in excavation will make 


Gravel or sand 

Clav 


1087 cubic yards 
1111 " 
1136 " 
1176 " 

714 " 

625 " 
measured in excavation 


920 cubic yards 

900 " 

880 " 

8.50 " 
1400 " 
1600 " 
of embaukmeut. 


Loam 

Loose vegetable soil 

Rock, larg(,' pieces 

*' small " 





97. Allowance for shrinkage. On account of the initial 
expansion and subsequent contraction of earth, it becomes 
necessary to form embankments higher than their required 
ultimate form in order to allow for the subsequent shrinkage. 
As the shrinkage appears to be all vertical (practically), the 
embankment must be formed as shown in Fi<2:. .58. Tlic effect 



114 



RAILROAD CONSTRUCTION. 



97. 



of shrinkage should not be confounded with that of sUpping of 
the sides, which is especially apt to occur if the embankment is 
subjected to heavy rains very soon after being formed, and also 
when the embankments are originally steep. It is often difficult 





Fig. 58. 



to form an embankment at a slope of 1 : 1 which will not slip 
more or less before it hardens. 

Very high embankments shrink a greater percentage than 
lower ones. Various rules giving the relation between shrink- 
age and height have been suggested, but they vary as badly as 
the suggested coefficients of contraction, probably for the same 
causes. As the fact is unquestionable, however, the extra 
height of the embankment must be varied somewhat as in Fig. 
59, which represents a longitudinal section of an embankment. 




Fig. 59. 

As considerable time generally elapses between the completion 
of the embankment and the actual riinnincr of trains, the o^rade 
ad will generally be nearly flattened down to its ultimate form 
before traffic commences, but such grades are occasionally objec- 
tionable if added to what is already a ruling grade. With some 
kinds of soil the time required for complete settlement may be 
as much as two or three years, but, even in such cases, it is 



§ 98. EARTHWORK. 115 

probable that one-half of the settlement will take place during 
the first six months. The engineer should therefore require 
the contractor to make all fills about 8 to 15^ (accordin*^ to 
the material) higher than the profiles call for, in order that 
subsequent shrinkage may not reduce it to less than the re- 
quired volume. 

98. Methods of forming embankments. When the method is 
not otherwise ol>jectionable, a high embankment can be formed 
very cheaply (assuming that carts or wheelbarrows are used) by 
dumjiing over the. end and building to the full height (or even 
higher, to allow for shrinkage) as the embankment proceeds. 
This allows more time for shrinkage, saves nearly all the cost of 
spreading (see Item 4, § 111), and reduces the cost of roadways 
(Item 5). Of course this method is especially applicable when 
the material comes from a place as liigh as or higher than grade, 
so that no up-hill hauling is required. 

Another method is to spread it in layers two or three feet 
thick (see Fig. GO), which are made concave upwards to avoid 

limiHUUiiimmmimiiimmuiiniv 




Fig. 60. 

possible sliding on each other. Spreading in layers has the 
advantage of partially ramming each layer, so that the subse- 
quent shrinkage is very small. Sometimes small trendies are 
dug along the lines of the toes of the embankment. This will 
frequently prevent the sliding of a large mass of the embank- 
ment, which will then require extensive and costly repairs, to 
say nothing of possible accidents if the sliding occurs after the 
road is in operation. Incidentally these trenches will be of 
value in draining the subsoil. AVhen circumstances require an 
embankment on a hillside, it is advisable to cut out "steps" to 
prevent a possible sliding of the whole embankment. Merely 



116 RAILROAD CONSTRUCTION. § 99. 

ploughing the side-hill will often be a cheajDer and sufficiently 
effective method. 




Fig. 61. 



Occasionally the formation of a very high and lono- embank- 
ment may be most easily and cheaply accomplished by building 
a trestle to grade and opening the road. Earth can then be 
procured where most convenient, perhaps several miles away, 
loaded on cars with a steam-shovel, hauled by the trainload, and 
dumped from the cars with a patent unloader. On such a large 
scale, the cost per yard would be very much less than by ordi- 
nary methods — enough less sometimes to more than pay for the 
temporary trestle, besides allowing the road to be opened for 
traffic very much earlier, which is often a matter of prime 
financial importance. It may also obviate the necessity for 
extensive borrow-pits in the immediate neighborhood of the 
heavy fill and also utilize material which would otherwise be 
wasted. 



COMPUTATION OF HAUL. 



99. Nature of subject. As will be shown later when analyz- 
ing the cost of earthwork, the most variable item of cost is that 
depending on the distance hauled. As it is manifestly imprac- 
ticable to calculate the exact distance to which every individual 
cartload of earth has been moved, it becomes necessary to devise 
a means which will give at least an equivalent of the haulao-e of 
all the earth moved. Evidently the average haul for any mass 
of earth moved is equal to the distance from the center of 2:rav- 
ity of the excavation to the center of gravity of the embank- 



§100. 



EARTHWORK. 



117 



merit formed by the excavated material. As a rongli approxi- 
mation the center of gravity of a cut (or fill) may sometimes be 
considered to coincide with the center of gravity of that part of 
tlie profile representing it, but the error is frequently very large. 
The center of gravity may be determined by various methods, 
but the method of the " mass diagram " accomplishes the same 
ultimate purpose (the determination of the haul) with all-suffi- 
cient accuracy and also furnishes other valuable information. 

100. Mass diagram. In Fig. 62 let A' R . . . G' represent 
a profile and grade line drawn to the usual scales. Assume A' 




Fig. 63.— Mass Diagram. 

to be a point past which no earthwork will be hauled. Above 
every station point in the profile draw an ordinate which 
will represent the algebraic sum of the cubic yards of cut and 
fill (calling cut + and fill — ) from the point A' to the point 
considered. In doing this shrinkage must be allowed for by 
considering how much embankment would actually be made by 
so many cubic yards of excavation of such material. For 
example, it will be found that 1000 cubic yards of sand or 
gravel, measured in place (see § 97), will make about 920 cubic 
yards of embankment; therefore all cuttings in sand or gravel 
should be discounted in about this proportion. Excavations in 
rock should be increased in the proper ratio. In short, all ex- 
cavations should be valued according to the amount of settled 
embankment that could be made from them. The computations 
may be made systematically as shown in the tabular form. Place 



118 



RAILROAD CONSTRUCTION. 



101. 



in the first column a list of the stations; in the second column, 
the number of cubic yards of cut or fill between each station 
and the preceding station ; in the third and fourth columns, the 
kind of material and the proper shrinkage factor; in the fifth 
column, a repetition of the quantities in cubic yards, except that 
the excavations are diminished (or increased, in the case of rock) 
to the number of cubic yards of settled embankment which may 
be made from tliem. In the sixth column, place the algehraic 
sum of the quantities in the fifth column (calling cuts + and 
fills — ) from the starting-point to the station considered. These 
algebraic sums at each station will be the ordinates, drawn to 
some scale, of the mass curve. The scale to be used will depend 
somewhat on whether the work is heavy or light, but for ordi- 
nary cases a scale of 5000 cubic yards per inch may be used. 
Drawing these ordinates to scale, a curve A^ B^ . . . G may be 
obtained by joining the extremities of the ordinates. 



Sta. 


Yards{-* + 


Material. 


Shrinkage 
factor. 


Yards, reduced 
for shrinkage. 


Ordinate in 
mass curve. 


46 +70 

47 

48 

+ 60 
49 
50 
51 
32 

+ 30 
53 

+ 70 
54 

+ 42 
55 
56 
57 












+ 175 

+ 1788 
+ 2341 
+ 2198 
+ 1292 

- 693 
-2414 

- 2526 
-2243 

- 1954 
-2006 
-2077 
-1828 

- 710 
+ 462 


+ 195 
+ 1792 
+ 614 

- 143 

- 906 
-1985 
-1721 

- 112 
+ 177 
+ 180 

- 52 

- 71 
+ 276 
+ 1242 
+ 1302 


Clayey soil 


— 10 per ceut 

-10 

-10 


+ 175 
+ 1613 
+ 553 

- 143 

- 906 
-1985 
-1721 

- 112 
+ 283 
+ 289 

- 52 

- 71 
+ 249 
+ 1118 
+ 1172 


















Hard rock 
1 < (« 


+60 per cent 
+60 " 






Clayey soil 
i i It 

<( 1 < 


— 10 percent 
-10 " 
-10 



101. Properties of the mass curve. 

1. The curve will be rising while over cuts and falling 
while over fills. 

2. A tangent to the curve will be horizontal (as at B^ 7), E^ 
F^ and G) when passing from cut to fill or from fill to cut. 



§ 101. EARTHWORK. 119 

3. Wlieii the curve is helow the " zero line " it shows that 
material must be drawn backward (to the left) ; and vice versa^ 
when the curve is above the zero line it shows that material 
must be drawn ybri^arc? (to the right). 

4. When the curve crosses the zero line (as at A and 6') it 
shows (in this instance) that the cut between A' and B' will just 
provide the materinl required for the fill between^' and 6^', and 
that no material should be hauled past C\ or, in general, past 
any intersection of the mass curve and the zero line. 

5. If any horizontal line be drawn (as ah)^ it indicates that 
the cut and fill between a' and b' will just balance. 

6. When the center of gravity of a given volume of 
material is to be moved a given distance, it makes no difference 
(at least theoretically) how far each individual load may be 
hauled or how any individual load may be disposed of. The 
summation of the products of each load times the distance 
hauled will be a constant, whatever the method, and will equal 
the total volume times the movement of the center of gravity. 
The a/verage haul, which is the movement of the center of 
gravity, will therefore equal the summation of these products 
divided by the total volume. If we draw two horizontal ])ar- 
allel lines at an infinitesimal distance dx a})art, as at «/>, the 
small increment of cut dx at a' will fill the corresponding incre- 
ment of fill at b\ and this material must be hauled the distance 
ab. Therefore the product of ab and dx, which is the product 
of distance times volume, is represented by the area of the 
infinitesimal rectangle at ab, and the total area ABC represents 
the summation of volume times distance for all the earth move- 
ment between A' and C . This sunnnation of ])roducts divided 
by the total volume gives the average haul. 

7. The horizontal line, tangent at E and cutting the curve 
at e,f, and g, shows that the cut and fill between e' and E' will 
just balance, and that a possible method of hauling (whether 
desirable or not) would be to 'M>orrow" earth for the fill 
between C and e' , use the material between D' and K' foi- the 



1^0 RAILROAD CONSTRUCTION. § 101. 

fill between e and I)\ and similarly balance cut and fill between 
E' andy^ and also between y^ and g' . 

8. Similarly the horizontal line hldw, may be drawn cuttino- 
the curve, which will show another possiUe method of hauling. 
According to this plan, the fill between C and h' would be 
made by borrowing ; the cut and fill between h' and h' would 
balance; also that between Jc' and V and between V and m' . 
Since the area ehDkE represents the measure of haul for tlie 
earth between e and E\ and the other areas measure the corre- 
sponding hauls similarly, it is evident that the sum of the areas 
eliDhE and ElFmf^ which is the measure of haul of all the 
material between e' and/', is largely in excess of the sum of 
the areas IWk^ hEl^ and IFm^ plus the somewhat uncertain 
measures of haul due to borrowing material for e'h' and wastino- 
the material between m and/'. Therefore to make the meas- 
ure of haul a minimum a line should be drawn which will 
make the sum of the areas between it and the mass curve a 
minimum. Of course this is not necessarily the cheapest plan, 
as it implies more or less borrowing and wasting of material, 
which may cost more than the amount saved in haul. The 
comparison of the two methods is quite simple, however. Since 
the amount of fill between e and li' is represented by the differ- 
ence of the ordinates at e and A, and similarly for m' and/', it 
follows that the amount to be borrowed between e' and li will 
exactly equal the amount wasted between in and /'. By the 
first of the above methods the haul is excessive, but is definitely 
known from the mass diagram, and all of the material is util- 
ized ; by the second method the haul is reduced to about one- 
half, but there is a known quantity in cubic yards wasted at one 
place and the same quantity borrowed at another. The leno>th 
of haul necessary for the borrowed material would need to be 
ascertained ; also the haul necessary to w^aste the other material 
at a place where it would be unobjectionable. Frequently this 
is best done by widening an embankment beyond its necessary 
width. The computation of the relative cost of the above 
methods will be discussed later (§ 116). 



§ 102. EARTHWORK. 121 

9. Suppose that it were deemed best, after drawing the mass 
curve, to introduce a trestle between s' and v' . tlms savin (»• an 
amount in fill equal to tv. If such Lad been tlie original desi<ni 
the mass curve would have been a straight horizontal line 
between s and i and would continue as a curve which would be 
at all points a distance tv above the curve vFmzfGg. If the 
line Ef is to be used as a zero line, its intersection with the new 
curve at x will show that the material between E' and z' will 
just balance if the trestle is used, and that the amount of liaul 
will be measured by the area between the line Ex and the broken 
line Estx. The same computed result may. be obtained without 
drawing the auxiliary curve txn ... by drawing the horizontal 
line zy at a distance xz {j=^ tv) below Ex. The amount of the 
haul can then be obtained by adding the triangular area between 
Es and the horizontal line Ex, the rectangle between st and Ex^ 
and the irregular area between vFz and y . . . z (which last is 
evidently equal to the area between tx and E . . . x). The dis- 
posal of the material at the right of z' would then be governed 
by the indications of the profile and mass diagram wdiicli would 
be found at the right of g' . In fact it is difficult to decide with 
the best of judgment as to the proper disposal of material with- 
out having a mass diagram extending to a considerable distance 
each side of that part of the road under immediate considera- 
tion . 

102. Area of the mass curve. The area may be computed 
most readily by means of a planimeter, -which is capable with 
reasonable care of measuring such areas with as great accuracy 
as is necessary for this work. If no such instrument is obtain- 
able, the area may be obtained by an application of '^ Simpson's 
rule." The ordinates will usually be spaced 100 feet apart. 
Select an even number of such spaces, leaving, if necessary, one 
or more triangles or trapezoids at the ends for separate and 
independent computation. Let y^ - • • y,x ^^e the ordinates, i.e., 
the number of cubic yards at each station of the mass curve., or 
the figures of "column six" referred to in § 100. Let the 
uniform distance between ordinates (= 100 feet) be called 1, i.e., 



122 RAILROAD COISSTRUCTION, % 103. 

one station. Then tlie units of the resulting area will be cubic 
yards hauled one station. Then the 

Area = ^[y, + ^{y, -|- ^3 + . . . y^^^ _ ^p _|_ ^^y^ + 2/4 + . . . y^,^_^^) + y^l (70) 

When an ordinate occurs at a substation, tlie best plan is to 
ignore it at first and calculate the area as above. Then, if the 
difference involved is too great to be neglected, calculate the 
area of the triangle having the extremity of the ordinate at the 
substation as an apex, and the extremities of the ordinates at the 
adjacent stations as the ends of the base. This may be done by 
finding the ordinate at the substation tliat would be a propor- 
tional between the ordinates at the adjacent full stations. Sub- 
tract this from the real ordinate (or vice versa) and multiply the 
difference by i X 1. An inspection will often show that the 
correction thus obtained would be too small to be worthy of con- 
sideration. If there is more than one substation between two 
full stations, the corrective area will consist of two triangles and 
one or more trapezoids which may be similarly computed, if 
necessary. 

When the zero line (Fig. 62) is shifted to eE^ the drop from 
AC (produced) to E is known in the same units, cubic yards. 
This constant may be subtracted from the numbers (" column 
4," § 100) representing the ordinates, and will thus give, with- 
out any scaling from the diagram, the exact value of the modi- 
fied ordinate?. 

103. Value of the mass diagram. The great value of the mass 
diagram lies in the readiness with which different plans for the 
disposal of material may be examined and compared. When 
the mass curve is once drawn, it will generally require only a 
shifting of the horizontal line to show the disposal of the material 
by any proposed method. The mass diagram also shows the 
extreme length of haul that will be required by any proposed 
method of disposal of material. This brings into consideration 
the "limit of profitable haul," which will be fully discussed in 
§ 116. For the present it may be said that with each method 
of carrying material there is some limit beyond which the expense 



§ 104. EARTHWORK. 123 

of hauling will exceed the loss resulting from borrowing and 
wasting. With wheelbarrows and scrapers the limit of profit- 
able haul is comparatively short, with carts and tram-cars it is 
much longer, while with locomotives and cars it may be several 
miles. If, in Fig. 62, eE ov ^J^ exceeds the limit of profitable 
haul, it shows at once that some such line as hklm should be 
drawn and the material disposed of accordingly. 

104. Changing the grade line. The formation of the mass 
curve and the resulting plans as to the disposal of material are 
based on the mutual relations of the grade line and the surface 
profile and the amounts of cut and fill which are thereby im- 
plied. If the grade line is altered, every cross-section is 
altered, the amount of cut and fill is altered, and the mass 
curve is also changed. At the farther limit of the actual 
change of the grade line the revised mass curve will have (in 
general) a different ordinate from the previous ordinate at that 
point. From that point on, the revised mass curve will be par- 
allel to its former position, and the revised curve may be treated 
similarly to the case previously mentioned in which a trestle was 
introduced. Since it involves tedious calculations to determine 
accurately how much the volume of earthwork is altered by a 
change in grade line, especially through irregular country, the 
effect on the mass curve of a change in the grade line cannot 
therefore be readily determined except in an approximate way. 
liaising the grade line will evidently increase the fills and 
diminish the cuts, and vice versa. Therefore if the mass curve 
indicated, for example, either an excessiv^ely long haul or the 
necessity for borrowing material (implying a fill) and wasting 
material farther on (implying a cut), it would be possible to 
diminish the fill (and hence the amount of material to l)e bor- 
rowed) by lowering the grade line near that place, and diminish 
the cut (and hence the amount of material to be wav'^ted) by 
raising the grade line at or near the ])lace farther on. Whether 
the advantage thus gained would compensate «for the possibly 
injurious effect of these changes on the grade line would require 
patient investigation. But the method outlined shows how the 



124 



ItAILROAD CONSTRUCTION. 



105. 



mass curve might be used to indicate a possible change in -rade 
line which might be demonstrated to be profitable. ^ 

105. Limit of free haul. It is sometimes specified in con- 
tracts for earthwork that all material shall be entitled to free 
haul up to some specified limit, say 500 or 1000 feet, and that 
all material drawn farther than that shall be entitled to an 
allowance on the excess of distance. It is manifestly imprac- 
ticable to measure the excess for each load, as much so as to 
measure the actual haul of each load. The mass diagram also 
solves this problem very readily. Let Fig. 63 represent a pro- 




Fig. G3. 



file and mass diagram of about 2000 feet of road, and suppose 
that 800 feet is taken as the limit of free haul. Find two 
points, a and h, in the mass curve which are on the same hori- 
zontal Une and which are 800 feet apart. Project these points 
down to a' and V. Then the cut and fill between a' and h' will 
just balance, and the cut between A' and a' will be needed for 
the fill between h' and C. In the mass curve, the area between 
the horizontal line ah and the curve aBh represents the haula-e 
of the material between a' and V , which is all free. The reel- 
angle ahnn represents the haulage of the material in the cut 
A'a' across the 800 feet from a' to h\ This is also free. The 
sum of the two areas Aam and hiC represents the haulage 
entitled to an allowance, since it is the summation of the products 
of cubic yards times the excess of distance hauled. 



§ 105. EARTUWORK. 125 

If tlie amount of cut and lill was synnnetrical about the 
point B\ the mass curve would be a symmetrical curve about the 
vertical line through /i, and the two limiting lines of free haul 
would be placed symmetrically about B and B' . h\ o-eneral 
there is no such symmetry, and frequently the difference is con- 
siderable. The area ciBhnm will be materially chano-ed accord- 
ing as the two vertical lines am and hi, always 800 feet apart, 
are shifted to the right or left. It is easy to show^ that the area 
aBhnm is a maximuni when ah is horizontal. The minimum 
value would be obtained either when 7)i reached A or n reached 
C, depending on the exact form of the curve. Since the posi- 
tion for the minimum value is manifestly unfair, the best dehnite 
value obtainable is the maximum, which must be obtained as 
above described. Since aBhim, is made maximum, the re- 
mainder of the area, which is the allow^ance for overhaul, be- 
comes a minhnum. The areas Aam and ICn may be obtained 
as in § 102. If the whole area AaBhCA has been previously 
computed, it may be more convenient to compute the area 
aBhnm and subtract it from the total area. 

Since the intersections of the mass curve and the " zero line " 
mark limits past wdiich no material is drawm, it follows that 
there will be no allowance for overhaul except where the dis- 
tance between consecutive intersections of the zero line and mass 
curve exceeds the limit of free haul. 

Frequently all allowances for overhaul are disregarded ; the 
profiles, estimates of quantities, and the required disposal of ma- 
terial are shown to bidding contractors, and they must tlien make 
their own allowances and bid accordingly. This method luis 
the advantage of avoiding possible disputes as to the amount of 
the overhaul allow^ance, and is popular with railroad companies on 
this account. On the other hand the facility with which differ- 
ent plans for the disposal of material may be studied and com- 
pared by the mass-curve method facilitates the adoption of the 
most economical plan, and the elimination of uncertaintv will 
frequently lead to a safe reduction of the bid, and so the method 
is valuable to both the railroad company and the contractor. 



126 RAILROAD CONSTRUCTION, § 106. 

ELEMENTS OF THE COST OF EARTHWORK. 

(The following analysis of the cost of earthwork follows the 
general method given in the well-known papers published bj 
Ellwood Morris, C.E., in the Journal of the Franklin Institute 
in September and October, 1841. l^umerous corroborative 
data have been obtained from various other- sources, and also 
figures on methods not then in vogue.) 

106. General divisions of the subject. The variations in the 
cost of earthwork are caused by the greatly varying conditions 
under which the work is done, chief among which is character 
of material, method of carriage, and length of haul. Any gen- 
eral system of computation must therefore differentiate the total 
cost into such elementary items that all differences due to varia- 
tions in conditions may be allowed for. The variations due to 
character of material will be allowed for by an estimate on loose 
light sandy soil, and also an estimate on the heaviest soils, such 
as stiff clay and hard-pan. These represent the extremes (ex- 
cluding rock, which will be treated separately), and the cost of 
intermediate grades must be estimated by interpolating between 
the extreme values. The general divisions of the subject will 
be:* 

1. Loosening. 

2. Loading. 

3. Hauling. 

4. Spreading. 

5. Keeping roadways in order. 

6. Kepairs, wear, depreciation, and interest on cost of plant. 

7. Superintendence and incidentals. 

8. Contractor's profit. 

By making the estimates on the basis of $1 per day for the 
cost of common labor, it is a simple matter to revise the esti- 
mates according to the local price of labor by multiplying the 
final estimate of cost by the price of labor in dollars per day. 

* Trautwine. 



§ 107. EAHTIIWORK. \2ll 

107. Item 1. Loosening, (a) Ploughs. Very liglit sandy 
soils can frequently be shovelled without any previous loosenin*'- 
but It is generally economical, even with very light material, to 
use a plough. Morris quotes, as the results of experiments, 
that a three-horse plough would loosen from 250 to 800 cubic 
yards of earth per day, which at a valuation of So per day 
would make the cost per yard vary from 2 cents to O.G cent. 
Trautwine estimates the cost on the basis of two men handlino- 
a two-horse plough at a total cost of $3.87 per day, beino- ,^l 
each for the men, 75 c. for each horse, and an allowance of 37 c. 
for the plough, harness, etc. From 200 to 600 cubic yards is 
estimated as a fair day's work, which makes a cost of 1.9 c. to 
0.65 c. per yard, which is substantially the same estimate as 
above. Extremely heavy soils have sometimes been loosened 
by means of special ploughs operated by traction-engines. 

(b) Picks. When picks are used for loosening the earth, as 
is frequently necessary and as is often done when plou^hino- 
would perhaps be really cheaper, an estimate - for a fair day's 
work is from 14 to 60 cubic yards, the 14 yards being the esti- 
mate for stiff clay or cemented gravel, and the 60 yards the esti- 
mate for the lightest soil that would require loosening. At 81 
per day this means about 7 c. to 1.7 c. per cubic yard, which is 
about three times the cost of ploughing. Five feet of the face 
is estimated f as the least width along the face of a bank that 
should be allowed to enable each laborer to work with freedom 
and hence economically. 

(c) Blasting. Although some of the softer shaly rocks may 
be loosened with a pick for about 15 to 20 c. per yard, yet rock 
in general, frozen earth, and sometimes even compact clay is 
most economically loosened by blasting. The subject of blast- 
ing will be taken up later, §g 117-123. 

(d) Steam-shovels. The items of loosening and loading merge 
together with this method, which will therefore be treated in 
the next section. 

* Trautwine. f Hurst. 



128 RAILROAD CONSTRUCTION. § 108. 

108. Item 2. Loading, (a) Hand-shovelling. Much depends 
on proper management, so that the shovellers need not wait 
unduly either for material or carts. With the best of manage- 
ment considerable time is thus lost, and yet the intervals of rest 
need not be considered as entirely lost, as it enables the men to 
work, while actually loading, at a rate which it would be physi- 
cally impossible for them to maintain for ten hours. Seven 
shovellers are sometimes allowed for each cart ; otherwise there 
should be five, two on each side and one in the rear. Economy 
requires that the number of loads per cart per day should be 
made as large as possible, and it is therefore wise to employ as 
many shovellers as can work without mutual interference and 
without wasting time in waiting for material or carts. The 
figures obtainable for the cost of this item are unsatisfactory on 
account of their large disagreements. The following are quoted 
as the number of cubic yards that can be loaded into a cart by 
an average laborer in a working day of ten hours, the lower 
estimate referring to heavy soils, and the higher to light sandy 
soils : 10 to 14 cubic yards (Morris), 12 to 17 cubic yards (Has- 
koll), 18 to 22 cubic yards (Hurst), IT to 24 cubic yards (Traut- 
wine), 16 to 48 cubic yards (Ancelin). As these estimates are 
generally claimed to be based on actual experience, the discre- 
pancies are probably due to difierences of management. If the 
average of 15 to 25 cubic yards be accepted, it means, on the 
basis of $1 per day, 6.7 c. to 4c. per cubic yard. These esti- 
mates apply only to earth. Bockworic costs more, not only 
because it is harder to handle, but because a cubic yard of solid 
rock, measured in place, occupies about 1.8 cubic yards when 
broken up, while a cubic yard of earth will occupy about 1.2 
cubic yards. Eockwork will therefore require about 50^ more 
loads to haul a given volume, measured in place, than will the 
same nominal volume of earthwork. The above authorities give 
estimates for loading rock varying from 6.9 c. to 10 c. per cubic 
yard. The above estimates apply only to the loading of carts 
or cars with shovels or by hand (loading masses of roek). The 



§ 108. EARTHWORK. 120 

cost of loading wheelbarrows and the cost of scraper work will 
be treated under the item of hauling. 

(b) Steam-shovels.- AVhenever the magnitude of the work 
will warrant it there is- great economy in the use of steam-shovels. 
These have a "bucket" or "dipper" on the end of a long 
beam, the bucket having a capacity varying from ^ to 2^ cubic 
3'ards. Steam-shovels handle all kinds of material from the 
softest earth to shale rock, earthy material containing large 
boulders, tree-stumps, etc. The capacity of the lai-ger sizes is 
about 3000 cubic- yards in 10 hours. They perform all the 
work of loosening and loading. Their economical working 
requires that the material shall be hauled away as fast as it can 
be loaded, wdiich usually means that cars on a track, hauled by 
horses or mules, or still better by a locomotive, shall be used. 
The expenses for a steam-shovel, costing about $5000, will 
average about $1000 per month. Of this the engineer will get 
$100 ; the fireman $50 ; the cranesman 890 ; repairs perhaps 
§250 to 8300; coal, from 15 to 25 tons, cost very variable on 
account of expensive hauling; water, a very uncertain amount, 
sometimes costing 8100 per month; about five laborers and a 
foreman, the laborers getting 81.25 per day and the foreman 
82.50 per day, which will amount to 8227.50 per month. 
This gang of laborers is employed in shifting the shovel when 
necessary, taking up and relaying tracks foi* the cars, shifting 
loaded and unloaded cars, etc. In shovelling through a deep 
cut, the shovel is operated so as to undermine the upper parts 
of the cut, which then fall down within reach of the shovel, thus 
increasing the amount of material handled for each new position 
of the shovel. If the material is too tough to fall down by its 
own weight, it is sometimes found economical to employ a gang 
of men to loosen it or even blast it rather than shift tlie shovel 
so frequently. Non-condensing engines of 50 horse-power use 
so much water that the cost of water-supply becomes a serious 

* For a thorough treatment of the capabilities, cost, and raanacrPTnent of 
;team-shovels the reader is referred to "Steam-shovels and Steam-shovel 
Work," by E. A. Hermann. D. Van Nostrand Co., New York. 



130 RAILROAD CONSTRUCTION. % 109. 

matter if water is not readily obtainable. The lack of water 
facilities will often justify the construction of a pipe line from 
some distant source and the installation of a steam-pump. 
Hence the seemingly large estimate of $100 per month for 
water-supply, although under favorable circumstances the cost 
may almost vanish. The larger steam-shovels wdll consume 
nearly a ton of coal per day of 10 hours. The expense of haul- 
ing this coal from the nearest railroad or canal to the location of 
the cut is often a very serious item of expense and may easily 
double the cost per ton. Some steam-shovels have been con- 
structed to be operated by electricity obtained from a plant 
perhaps several miles away. Such a method is especially 
advantageous when fuel and water are difficult to obtain. 

109. Item 3. Hauling. The cost of hauling depends on the 
number of round trips per day that can be made by each vehicle 
employed. As the cost of each vehicle is practically the same 
Avhether it makes many trips or few, it becomes important that 
tlie number of trips should be made a maximum, and to that 
end there should be as little delay as possible in loading and un- 
loading. Therefore devices for facilitating the passage of the 
vehicles have a real money value. 

(a) Carts. The average speed of a horse hauling a two- 
wheeled cart has been found to be 200 feet per minute, a little 
slower when hauling the load and a little faster when returning 
empty. This figure has been repeatedly verified. It means an 
allowance of one minute for each 100 feet (or "station") of 
*'lead — the lead being the distance the earth is hauled." The 
time lost in loading, dumping, waiting to load, etc., has been 
found to average 4 minutes per load. Representing the num- 
ber of stations (100 feet) of lead by 5, the number of loads 
handled in 10 hours (600 minutes) would be 600 — {s-\- 1). The 
number of loads per cubic yard, measured in the bank, is differ- 
entiated by Morris into three classes, viz. : 

3 loads per cubic yard in descending hauling ; 
3^ " " " " " level hauling ; and 

4 " " " *' " ascending hauling. 



§109. EARTUWORK. 131 

Attempts have been made to estimate the effect of the grade 
of the roadway by a theoretical consideration of its rate, and of 
the comparative strength of a horse on a level and on various 
grades. While such computations are always practicable on a 
railway (even on a temporary construction track), the traction on 
a temporary earth roadway is always very large and so very 
variable that any refinements are useless. On railroad earth- 
work the hauling is generally nearly level or it is descending — 
forming embankments on low ground with material from cuts in 
high ground. The only common exception occurs when an 
embankment is formed from borrow-pits on low ground. One 
method of allowing for ascending grade is to add to the hori- 
zontal distance 14 times the difference of elevation for work 
with carts and 24 times the difference of elevation for work 
with wheelbarrows, and use that as the lead. For example, 
using carts, if the lead is 300 feet and there is a difference of 
elevation of 20 feet, the lead would be considered equivalent to 
300 + (14 X 20) = 580 feet on a level. 

Trautwine assumes the average load for all classes of work 
to be \ cubic yard, which figure is justified by large experience. 
Using one figure for all classes of work simplifies the calculations 
and gives the number of cubic yards carried per day of 10 hours 

equal to r^ — ; — tt. Dividino: the cost of a cart per dav by the 

^3(^ + 4) ^ ir . J 

number of cubic yards carried gives the cost of hauling per 
yard. In computing the cost of a cart per day, Trautwine 
refers to the practice of having one driver manage four carts, 
thus making a charge of 25 c. per day for each cart for the 
driver. 75 c. is allowed for the horse, which is supposed to be 
the total cost, including that for Sundays and rainy days. 25 c. 
more is allowed for the cart, harness, repairs, etc., thus making 
a total cost of $1.25 per day. Some contractors employ a 
greater number of drivers and expect each to assist in loading. 
There is found to be no saving in total cost per yard, while the 
ehances of loafing are perhaps greater. Morris instances five 
actual cases in which the cost of the cart (reduced to the basis of 



132 RAILROAD CONSTRUCTION. % 109. 

$1 per day for labor) varied from $1.37 to $1.48. The items 
of these costs were not given. 

Since the time required for loading loose rock is greater than 
for earthwork, less loads will be hauled per day. The time 
allowance for loading, etc., is estimated by Trautwine as 6 
minutes instead of 4 as for earth. Considering the great ex- 
pansion of rock when broken up (see § 97), one cubic yard of 
solid rock, measured in place, w^ould furnish the equivalent of 
five loads of earthwork of J cubic yard. Therefore, on the 
basis of five loads per cubic yard, the number of cubic yards 

handled per day per cart would be . 

r. . ^ . , 125 X ^(s + 6) 

(Jost per yard m cents = ' ' — — -. . (71) 

(b) Wagons. For longer leads (i.e., from J- to f of a mile) 
wagons drawn by two horses have been found most economical. 
The wagons have bottoms of loose thick narrow boards and are 
unloaded very easily and quickly by lifting the individual boards 
and breaking up the continuity of the bottom, thus depositing 
the load directly underneath the wagon. The capacity is about 
one cubic yard. The cost may be estimated on the same prin- 
ciples as that for carts. 

(c) Wheelbarrows. According to Trautwine, the speed of 
moving wheelbarrows may be considered the same as for carts, 
200 feet per minute ; the time spent in loading and dumping is 
li minutes, and in addition about Jg- of the time is wasted in 
short rests, adjusting the wheeling plants, etc. On the basis of 
$1 per day for labor, an allowance of 5 c. for the barrow, and 14 
loads per cubic yard, the cost of hauling per cubic yard (com- 
puted on the same principles as above) will be 

105 X 14(^ + 1.25) 

600 X 0.9 ^'^^) 



§ 109. EARTHWORK. . 138 

For rockwork the number of loads per cubic yard is estimated 
as 24, and the time spent in loading, etc., estimated at l.G min- 
utes instead of 1.25 minutes, which makes the estimate 

,. , 105x24(5 + 1.6) 
Cost per cubic yard = ^^^ ^ ^^^^^^ ^ . . ( ^ 3) 

(d) Scrapers. * Scrapers, or scoops, are especially useful in 
canal work, and also for railroad work when a low embankment 
is to be formed from borrow-pits at the sides, when the distance 
does not exceed 100 feet, nor the vertical height 15 feet. The 
slope should not exceed 1.5 to 1 . Under these conditions scraper 
work is cheaper than any other method. Scooping may be done 
• all in one direction, in which case two half -turns are made for 
each load moved ; or it may be done in both directions (from 
both sides on to a bank, or, in canal work, from the center to 
each bank), in w^hich case one load is hauled to each half -turn. 
The capacity of the scoops (the "drag" variety) is JL cubic 
yard ; the time lost in loading, unloading, and all other ways 
per load (except in turning) will average | minute ; the time lost 
in each half-turn (semi-circle) is J minute ; the speed of tlie 
horses may be estimated as 70 feet of lead per minute, the lead 
being here considered as the stem of the vertical and horizontal 
distances, and the estimate including: the time of o-oino- and re- 
turning. If a represents the sum of the horizontal and vertical 
distances, the number of cubic yards handled per day of 10 
hours by " side- scooping " will be 

/ 600 \ 

r. . [ \ . . -, 4200 



For ''double-scooping" the formula becomes 
/ 600 

Vto 



A -1 [ \ 1-1 ^ 4200 

0.1 rt which equals . 



* Condensed from Journ. Franklin Inst., Oct. 1841, by Morris. 



134 RAILROAD CONSTRUCTION. § 109. 

Dividing the cost of a scraper per day (estimated at 82.75) by 
the number of yards handled per day gives the average cost per 
vard. 

Except in very loose sandy soil it is best to plough the earth 
first, whicli will cost about 1 c. per yard. (See § 107.) Drag- 
scrapers are now made chiefly of steel, and their capacity is more 
nearly 0.15 cubic yard. AVheeled scrapers, having a capacitv 
of about 0.5 cubic yard, are frequently used with even greater 
economy and for greater distances, as they are cheaper than 
carts up to 250 or 300 feet of lead. Both drag- and wheel- 
scrapers are best operated in gangs of perhaps 10, using extra 
or " snap " teams to help load, and a few extra men to help in 
loading and unloading. The average cost of one scraper per 
day may thus be easily calculated and the average number of 
cubic yards handled per day computed as above, from which 
the cost per yard may be estimated. 

(e) Cars and horses. The items of cost by this method are 
{a) charge for horses employed, ijb) charge for men employed 
strictly in hauling, (c) charge for shifting rails when necessary, 
(rZ) repairs, depreciation, and interest on cost of cars and track. 
Part of this cost should strictly be classified under items 5 and 
6, mentioned in § 106, but it is perhaps more convenient to 
estimate them as follows. 

The traction of a car on rails is so very small and constant 
that grade resistance constitutes a very large part of" the total 
resistance if the grade is I'fc or more. For all ordinary grades 
it is sufiiciently accurate to say that the grade resistance is to 
the gross weight as the rise is to the distance. If the distance 
is supposed to be measured along the slope, the proportion is 
strictly true; i.e., on a Ifo grade the grade resistance is 1 lb. 
per 100 of weight or 20 lbs. per ton. If the resistance on a 
level at the usual velocity is yi^-, a grade of 1 : 120 (0.83^) will 
exactly double it. If the material is hauled down a grade of 
1 : 120, the cars will run by gravity after bein^ started. TJie 
work of hauling will then consist practically of hauling the 
empty cars up the grade. The grade resistance depends only 



§ 109. EARTinVORK. 135 

on the rate of grade and the weight, but the tractive resistance 
will be (J r eater per ton of lo eight for the unloaded than for the 
loaded cars. The tractive power of a horse is less on a grade 
than on a level, not only because' the horse raises his own weight 
in addition to the load, but is anatomically less capable of 
pulling on a grade than on a level. In general it will be pos- 
sible to plan the work so that loaded cars need not be hauled up 
a erade, unless an embankment is to be formed from a low 
borrow-pit, in which case another method would probably be 
advisable. These computations are chietly utilized in designing 
the method of work — the proportion of horses to cars. An 
example may be quoted from English practice (Hurst), in w^hich 
the cars had a capacity of 3J- cubic yards, weighing 30 cwt. 
empty. Two horses took live " wagons " f of a mile on a level 
railroad and made 15 journeys per day of 10 hours, i.e., they 
handled 250 yards per day. In addition to those on the 
^'straight road," another horse ^vas employed to make up the 
train of loaded wagons. With a short lead the straight-road 
horses were employed for this purpose. In the above example 
the number of men required to handle these cars, shift the 
tracks, etc., is not given, and so the exact cost of the above 
work cannot be analyzed. It may be noticed that the two 
horses travelled 22J miles per day, drawing in one direction a 
load, including the weight of the cars, of about 57,300 lbs. or 
28.65 net tons. Allowing yi^- as the necessary tractive force, 
it would require a pull of 477.5 lbs., or 239 lbs. for each horse. 
With a velocity of 220 feet per minute this would amount to 
\\ horse-power per horse, exerted for only a short time, how- 
ever, and allowing considerable time for rest and for drawing 
only the empty cars. The cars generally used in this country 
have a capacity of IJ cubic yards and cost about $65 apiece. 
Besides the shovellers and dumping-gang, several men and a 
foreman will be required to keep the track in order and to make 
the constant shifts that are necessary. Two trains are generally 
used, one of which is loaded while the other is run to the 
dump. Some passing-place is necessary, but this is generally 



136 RAILROAD CONSTRUCTION. % 109, 

provided by having a switch at the' cut and running the trains 
on each track alternately. This insures a train of cars always 
at the cut to keep tlie shovellers employed. The cost of haul- 
ing per cubic yard can only be computed when the number of 
laborers, cars, and horses employed are known, and these will 
depend on the lead, on the character of the excavation, on the 
grade, if any, etc., and must be so proportioned that the shovel- 
lers need not wait for cars to fill, nor the dumping-gang for 
material to handle, nor the horses and drivers for cars to haul. 
Much*skill is necessary to keep a large force in smooth running 
order. 

(f ) Cars and locomotives. 30-lb. rails are the lightest that 
should be used for this work, and 35- or 40-lb. rails are better. 
One or two narrow-gauge locomotives (depending on the length 
of haul), costing abont $2500 each, will be necessary to handle 
two trains of about 15 cars each, the cars having a .capacity of 
about 2 cubic yards and costing about 8100 each. Some cars 
can be obtained as low as $70. A force of about five men and 
a foreman will be required to shift the tracks. The track- 
shifters, except the foreman, may be common laborers. The 
dumping-gang will require about seven men. Even when the 
material is all taken down grade the grades may be too steep for 
the safe hauling of loaded cars down the grade, or for hauling 
empty cars up the grade. Under such circumstances temporary 
trestles are necessary to reduce the grade. ^Yhen these are 
used, the uprights and bracing are left in the embankment — 
only the stringers being removed. This is largely a necessity, 
but is partially compensated by the fact that the trestle forms a 
core to the embankment which prevents lateral shifting during 
settlement. The average speed of the trains may be taken as 
10 miles per hour or 5 miles of lead per hour. The time lost 
in loading and unloading is estimated (Trautwine) as 9 minutes 
or .15 of an hour. The number of trips per day of 10 hours 

^.jj ^ ^^^^ 10_ ^^ 50 "^ ^^ 

^ "1^ (miles of lead) -\- .15 (miles of lead) -\- .75* 
course this quotient Tnust be a whole number. Knowing the 



g 1 10. EARTUWOIiK. 137 

iiuinber of trains and their capacity, the total number of cubic 
yards handled is known, which, divided into the total daily cost 
of the trains, will give the cost of hauling per yard. The daily 
cost of a train will include 

(a) Wages of engineer, who frequently iires his own engine ; 

(h) Fuel, about J to 1 ton of bitumnious coal, depending on 
work done; 

(c) Water, a very variable item, fi-equently costing $3 to §5 
per day ; 

{(I) Repairs,' variable, frequently at rate of 50 to 60^ per 
year ; 

(e) Interest on cost and depreciation, 16 to 40;^. 

To these must be added, to obtain the total cost of the haul, 

(f) Wages of the gang employed in sliifting track. 

110. Choice of method of haul dependent on distance. 
In light side-hill work in which material need not be moved 
more than 12 or 15 feet, i.e., moved laterally across the road- 
bed, the earth may be moved most chea})ly by mere shovelling. 
Beyond 12 feet scrapers are more economical. At about 100 
feet drag-scrapers and wheelbarrows are equally economical. 
Between 100 and 200 feet wheelbarrows are generally cheaper 
than either, c'arts or drag-scrapers, but wheeled scrapers are 
always cheaper than wheelbarrows. Beyond 500 feet two- 
wheeled carts become the most economical up to about 1700 
feet ; then four-wheeled wagons become more economical u]) to 
3500 feet-. Beyond this cars on rails, drawn by horses or by 
locomotives, become cheaper. The economy of cars on rails 
becomes evident for distances as small as 300 feet provided the 
volume of the excavation w^ill justify the outlay. Locomotives 
will always be cheaper than horses and mules providing the 
work to be done is of sufhcient magnitude to justify the jnir- 
chase of the necessary plant and risk the loss in selling the plant 
ultimately as second-hand equipment, or keeping the plant on 
hand and idle for an indefinite ])eriod waiting for other work. 
Horses will not be economical for distances much over a mile. 
For greater distances locomotives are more economical, but the 



138 RAILROAD CONSTRUCTION. §111. 

question of "limit of profitable haul" (§ 116) must be closely 
studied, as the circumstances are certainly not common when it 
is advisable to haul material much over a mile. 

111. Item 4. Spreading. The cost of spreading varies with 
the method employed in dumping the load. When the earth is 
tipped over the edge of an embankment there is little if any 
necessary work. Trautwine allows about J c. per cubic yard 
for keeping the dumping-places clear and in order. This would 
represent the wages of one man at $1 ]3er day attending to the 
unloading of 1200 two- wheeled carts each carrying J cubic yard. 
1200 carts in 10 hours would mean an average of two per 
minute, which implies more rapid and efficient work than may be 
depended on. The allowance is probably too small. AYhen the 
material is dumped in layers some levelling is required, for 
which Trautwine allows 50 to 100 cubic yards as a fair day's 
work, costing from 1 to 2 cents per cubic yard. The cost of 
spreading will not ordinarily exceed this and is frequently noth- 
ing — all depending on the method of unloading. It should be 
noted that Mr. Morris's examples and computations (Jour. Frank- 
lin Inst., Sept. 181:1) disregard altogether any special charge 
for this item. 

112. Item 5. Keeping Roadways in order. This feature 
is important as a measure of true economy, whatever the system 
of transportation, but it is often neglected. A petty saving in 
such matters will cost many times as much in increased labor in 
haulino: and loss of time. With some methods of haul the cost 
is best combined with that of other items. 

(a) Wheelbarrows. Wheelbarrows should generally be run 
on planks laid on the ground. The adjusting and shifting of 
these planks is done by the wheelers, and the time for it is allowed 
for in the 10^ allowance for " short rests, adjusting the wheel- 
ing plank, etc." The actual cost of the planks must be added, 
but it would evidently be a very small addition per cubic yard 
in a large contract. When the wheelbarrows are run on planks 
placed on ' ' horses ' ' or on trestles the cost is very appreciable ; 
but the method is frequently used with great economy. The 



§ 114. EARTHWORK. 139 

variations in tlie requirements render any general estimate of 
SLich cost impracticable. 

(b) Carts and wagons. The cost of keeping roadways in 
order for carts and Avagons is sometimes estimated merely as so 
mncli per cubic yard, but it is evidently a function of the lead. 
The work consists in draining off puddles, filling up ruts, pick- 
ing up loose stones that may have fallen off the loads, and in 
general doing everything that will reduce the traction as much 
as possible. Temporary inclines, built to avoid excessive grade 
at some one point, are often measures of true economy. Traut- 
wine suggests ^l c. per cubic yard per 100 feet of lead for earth- 
work and y2_. c. for rockwork, as an estimate for this item when 
carts are used. 

(c) Cars. AVhen cars are used a sliifting-gang, consisting 
of a foreman and several men (say five), are constantly employed 
in shifting the track so that the material may be loaded and un- 
loaded where it is desired. The averao^e cost of this item mav 
be estimated by dividing the total daily cost of this gang by the 
number of cubic yards handled in one day. 

113. Item 6. Repairs, Wear, Depreciation, and Interest 
ON Cost of Plant. Tlie amount of this item evidently depends 
upon the character of the soil — the harder the soil the worse the 
wear and depreciation. The interest on cost depends on the 
current borrowing: value of monev. The estimate for this item 
has already been included in the allowances for horses, carts, 
ploughs, harness, wheelbarrows, steam-shovels, etc. Trautwine 
estimates J c. per cul)ic yard for picks and shovels. Deprecia- 
tion is generally a large percentage of the cost of earth-working 
tools, the life of all being limited to a few years, and of many 
tools to a few months. 

114. Item 7. Superintendence and Incidentals. The inci- 
dentals include water-carriers, trimming cuts to grade, digging 
the side ditches, trimming up the sides of borrow-pits to prevent 
their becoming unsightly, etc. These last operations yield but 
little earth and cost far more than the price paid per cubic yard. 
Morris allows 1 c. per cubic yard for this item ; Trautwine 



140 BAILROAD CONSTRUCTION. § 115. 

allows If to 2 c. for it; while others combine items 6 and 7 
and call them 5^ of the total cost, which method has the merit 
of making the cost of items 6 and 7 a function of the character 
of soil and length of lead. 

115. Items. Contractor's Profit. This is usually estimated 
at from 6 to 15^, according to the sharpness of the competition 
and the possible uncertainty as to true cost owing to unfavorable 
circumstances. The contractor's real profit may vary considerably 
from this. He often pays clerks, boards and lodges the laborers 
in shanties built for the purpose, or keeps a supply-store, and 
has various other items both of profit and expense. His profit 
is largely dependent on skill in so handUng the men that all can 
Avork effectively without interference or delays in w^aiting for 
others. An unusual season of bad weather will often affect the 
cost very seriously. It is a common occurrence to find that two 
contractors may be working on the same kind of material and 
under precisely similar conditions and at the same price, and yet 
one may be making money and the other losing it — all on ac- 
count of difference of management. 

116. Limit of profitable haul. As intimated in §§ 103 and 
110, there is with every method of haul a limit of distance be- 
yond w^liich the expense for excessive hauling will exceed the 
loss resulting from borrowing and wasting. Tliis distance is 
somewhat dependent on local conditions, thus requiring an inde- 
pendent solution for each particular case, but the general prin- 
ciples involved will be about as follows : Assume that it has been 
determined, as in Fig. 62, that the cut and fill will exactly bal- 
ance between two points, as between e and a?, assuming that, as 
indicated in § 101 (9), a trestle has been introduced between s 
and t^ thus altering the mass curve to Estxn . . . Since there 
is a balance between A' and (7', the material for the fill between 
C and (^ must be obtained either by " borrowing " in the im- 
mediate neighborhood or by transportation from the excavation 
between z and n' . If cut and fill have been approximately 
balanced in the selection of grade line, as is ordinarily done, 
borrowing material for the fill C'e' implies a wastage of material 



^116. EARTHWORK. 141 

at the cut z'n'. To compare the two methods, we may place 
against the phiii of borrowing and wasting, {a) cost, if any, of 
«xtra right of way that may be needed from whicli to obtain 
earth for the fill C'e'] (h) cost of loosening, loading, hauling 
a distance equal to that between the centers of gravity of the 
borrow-pit and of the fill, and the other expenses incidental to 
borrowing J/ cubic yards for the fill C"e' ; (c) cost of loosening, 
loading, hauling a distance equal to that between the centers 
of gravity of the cut z'?i' and of the spoil-bank, and the other 
expenses incidental to wasting J/ cubic yards at the cut z'7i' ; 
(d) cost, if any, of land needed for the spoil-bank. The cost of 
the other plan will be the cost of loosening, loading, hauling (tlie 
hauling being represented by the trapezoidal figure Gexn)^ and 
the other expenses incidental to making the fill C'e with the 
material from the cut z'n\ the amount of material being J/ cubic 
yards, which is represented in the figure by the vertical ordi- 
nate from e to the line Cn. The difference between these costs 
will be the cost, if any, of land for borrow-pit and spoil-bank 
plus the cost of loosening, loading, etc. (except hauling and 
roadways) of M cubic yards, minus the difference in cost of the 
excessive haul from Ce to xn and the comparatively short hauls 
from borrow-pit and to spoil- bank. 

As an illustration, taking some of the estimates previously 
given for operating with average material, the cost of all items, 
except hauling and roadways, would be about as follows : 
loosening, with plough, 1.2 c, loading 5.0 c, spreading 1.5 c., 
wear, depreciation, etc., .25 c, superintendence, etc., 1.5 c. ; 
total 8.95 c. Suppose that the haul for both borrowing and 
wasting averages 100 feet or 1 station. Then the cost of haul 
per yard, using carts, would be (§ 109, a) [125 X 3(1 -j- 4)] h- 
600 .= 3.125 c. The cost of roadways would be about 0.1 c. 
per yard, making a total of 3.225 c. per cubic yard. Assume 
M = 10000 cubic yards and the area Ce,rn = ISOOOO yards- 
stations or the equivalent of 10000 yards hauled 1800 feet. 
This haul would cost [125 X 3(18 + •!)] -^ 600 = 13.75 c. per 
cubic yard. The cost of roadways will be 1.^ X .1 or 1.8 c, 



142 



RAILROAD CONSTRUCTION. 



117. 



making a total of 15.55 c. for hauling and roadways. The 
difference of cost of hauling and roadways will be 15.55 — 
(2 X 3.225) = 9.10 c. per yard or 8910 for the 10000 yards. 
Oifsetting this is the cost of loosening, etc., 10000 yards, at 
8.95 c, costing $895. These figures may be better compared 
as follows : 



Long Haul. ■{ 



f Loosening, etc., 10000 yards, 
Hauling, " 10000 " 



8.95 c. 
15.55 c. 



I 



$ 895. 
1555. 

$2450. 



BORROWIXG 

AND Wasting. 



c. 1895. 
c. 895. 



i 



Loosening, etc., 10000 yards (borrowed), @ 8.95 

" 10000 " (wasted). @ 8.95 

Hauling, etc., 10000 " (borroAved), @ 3.225 c. 322.50 

10000 " (wasted), @ 3.225 c. 322.50 



$2485.00 



These costs are practically balanced, but no allowance has 
been made for right of way. If any considerable amount had 
to be paid for that, it would decide this particular case in favor 
of the long haul. This shows that under these conditions 1800 
feet is a^out the limit of profitable haul, the land costing nothing 
extra. 

BLASTING. 

117. Explosives. The effect of blasting is due to the ex- 
tremely rapid expansion of a gas which is developed by the 
decomposition of a very small amount of solid m.atter. Blasting 
compounds may be divided into two general classes, (a) slow- 
burning and (h) detonating. Gunpowder is a type of the slow- 
burning compounds. These are generally ignited by heat ; the 
ignition proceeds from grain to grain ; the heat and pressure 
produced are comparatively low. Xitro-glycerine is a type of 
the detonating compounds. They are exploded by a sliock 
which instantaneoicsly explodes the whole mass. The heat and 
pressure developed are far in excess of tliat produced by the 
explosion of poAvder. IN^itro-glycerine is so easily exploded 
that it is very da^ngerous to handle. It was discovered that if 
the nitro-glycerine was absorbed by a sjjongy material like infu- 



§ 117. EARTHWORK. 143 

serial earth, it was miicli less liable to explode, while its power 
when actually exploded was practically equal to ""ihat of the 
amount of pure nitro-glycerine contained in the dynamite, which 
is the name given to the mixture of nitro-glycerine and infusorial 
earth. Xitro-glycerine is expensive; many other explosive 
chemical compounds which properly belong to the slow-burning 
class are comparatively cheap. It has been conclusively demon- 
strated that a mixture of nitro-glycerine and some of the cheaper 
chemicals has a greater explosive force than the sum of the 
strengths of the component parts when exploded separately. 
AVhatever the reason, the fact seems established. The reason is 
possibly that the explosion of the nitro-glycerine is sufficiently 
powerful to produce a detonation of the other chemicals, which 
is impossible to produce by ordinary means, and that this explo- 
sion caused by detonation is more powerful than an ordinary 
explosion. The majority of the explosive compounds and 
" powders" on the market are of this character — a mixture of 
20 to 60 per cent, of nitro-glycerine with variable proportions 
of one or more of a great variety of explosive chemicals. 

The choice of the explosive depends on the character of the 
rock. A hard brittle rock is most effectively blasted by a 
detonating compound. The rapidity with which the full force 
of the explosive is developed has a shattering effect on a brittle 
substance. On the contrary, some of the softer tougher rocks 
and indurated clays are but little affected by dynamite. The 
result is but little more than an enlargement of the blast-hole. 
Quarrying must generally be done with blasting-powder, as the 
quicker explosives are too shattering. Although the results 
obtained by various experimenters are very variable, it may be 
said that pure nitro-glycerine is eight times as powerful as black 
powder, dynamite (75^ nitro-glycerine) six times, and gun-cotton 
four to six times as powerful. For open work where time is not 
particularly valuable, black powder is by far the cheapest, but 
in tunnel-headings, whose progress determines the progress of 
the whole work, dynamite is so much more effective and so 
expedites the work that its use becomes economical. 



144 



RAILROAD CONSTRUCTION. 



§118. 



118. Drilling. Altliougli many very com^^licated forms of 
drill-bars liave been devised, tlie best form (with slight modifi- 
cations to suit circumstances) is as shown in Fig. 64, (a) and (b). 





\ i^O 



Fig. r)4. 

The width should flare at the bottom {a) about 15 to 30^. For 
hard rock the curve of the edge should be somewhat flatter and 
for soft rock somewhat more curved than shown, Fig. 64, (ct). 
Sometimes the angle of the two faces is varied from that given. 
Fig. 64, (5), and occasionally tlie edge is purposely blunted so 
as to give a crushing rather than a cutting effect. The di'ills 
Avill require sharpening for each 6 to 18 inches deptli of hole, 
and will require a new edge to be worked every 2 to 4 days. 
For drilling vertical holes the churn-drill is the most econom- 
ical. The drill-bar is of iron, about 6 to 8 feet long, 1 J" in 
diameter, weighs about 25 to 30 lbs., and is shod with a piece 
of steel welded on. The bar is lifted a few inches between each 
blow, turned partially around, and allowed to fall, the impact 
doing the work. From 5 to 15 feet of holes, depending on the 
character of the rock, is a fair day's work — 10 hours. In very 
soft rocks even more tlian this may be done. This method is 
inapplicable for inclined holes or even for vertical holes in con- 
fined places, such as tunnel-headings. For such places the only 
practical hand method is to use hammers. This may be done 
by light drills and light hammers (one-man work), or by heavier 
drills held by one man and struck by one or two men with 
heavy hammers. The conclusion of an exhaustive investigation 
as to the relative economy of light or heavy hammers is that the 
light-hammer method is more economical for the softer rocks, 
the heavy-hammer method is more economical for the harder 



§119. 



EARTHWORK. 



145 



rucks, but that tlie liglit-liaiiuncr method is always more ex- 
peditious and hence to be preferred when time is important. 

The subject of macliine rock-drills is too vast to be treated 
here. The method is only practicable when the amount of 
work to be done is large, and especially when time is valuable. 
The machines are generally operated by compressed air for tun- 
nel-work, thus doing the additional service of supplying fresh 
air to the tunnel-headings wdiere it is most needed. The cost 
per foot of hole drilled is quite variable, but is usually some- 
what less than that of hand-drilling — sometimes but a small 
fraction of it. 

119. Position and direction of drill-holes. As the cost of 
drilling holes is the largest single item in the total cost of blast- 
ing, it is necessary that skill and judgment should be used in so 
locating the holes that the blasts will be most effective. The 
greatest effect of a blast will evidently be in the direction of the 
''line of least resistance." In a strictly homogeneous material 
this will be the shortest line from the center of the explosive to 
the surface. The variations in homogeneity on account of 
laminations and scams require that each case shall be judo-ed 
according to experience. In open-pit blasting it is generallv 
easy to obtain two and sometimes three exposed faces to the 
rock, making it a simple matter to drill holes so that a blast will 
do effective work. When a solid face of rock must be broken 
into, as in a tunnel-heading, the 
work is necessarily ineffectual and 
expensive. A conical or wedge- 
shaped mass will first be blown out 
by simultaneous blasts in the holes 
marked 1, Fig. %b\ blasts in the 
holes marked 2 and 3 will then com- 
plete the cross-section of the head- 
ing. A great saving in cost may 
often be secured by skilfully taking 
advantage of seams, breaks, and irregularities. When the work 
is economically done there is but little noise or throwing of rock, 

/ 




drill holes i.n tunne^l heading 
Fig. 65. 



146 BAILROAD CONSTRUCTION. § 120. 

a covering of old timbers and branches of trees generally sufficing 
to confine the smaller pieces which would otherwise fly up. 

120. Amount of explosive. The amount of explosive required 
varies as the cube of the line of least resistance. The best 
results are obtained when the line of least resistance is f of the 
depth of the hole ; also when the powder fills about i of the 
hole. For average rock the amount of j)owder required is as 
follows : 



Line of least resistance. 
Weight of powder 



2 ft. 



4 ft. 
2 lbs. 



6 ft. 



8 ft. 
16 lbs. 



Strict compliance with all of the above conditions would re- 
quire that the diameter of the hole should vary for every case. 
While this is impracticable, there should evidently be some 
variation in the size of the hole, depending on the work to be 
done. For example, a V hole, drilled 2' 8" deep, with its 
line of least resistance 2', and loaded with J lb. of powder, 
would be filled to a depth of ^'\ which is nearly i of the 
depth. A Z" hole, drilled 8' deep, with its line of least resist- 
ance 6', and loaded with Gf lbs. of powder, would be filled ta 
a depth of over 28'', which is also nearly ^ of the depth. One 
pound of blasting-powder will occupy about 28 cubic inches. 
Quarrying necessitates the use of nuuierous and sometimes 
repeated hght charges of powder, as a heavy blast or a powerful 
explosive like dynamite is apt to shatter the rock. This 
requires more powder to the cubic yard than blasting for mere 
excavation, which may usually be done by the use of J to i lb. 
of powder per cubic yard of easy open blasting. On account 
of the great resistance offered by rock when blasted in headings 
in tunnels, the powder used per cubic yard will run up to 2, 4 
and even 6 lbs. per cubic yard. As before stated, nitro- 
glycerine is about eight times (and dynamite about six times) as 
powerful as the same weight of powder. 

121. Tamping. Blasting-powder and the slow-burnino- ex- 
plosives require thorough tamping. Clay is probably the best^ 



§ 123 EARTUWOUK. 147 

but sand and fine powdered rock are also used. AVooden plugs, 
inverted expansiv^e cones, etc., are periodically reinvented by 
enthusiastic inventors, only to be discarded for the simpler 
methods. Owing to the extreme rapidity of the development 
of the force of a nitro-glycerine or dynamite explosion, tamping 
is not 60 essential with these explosives, although it unquestion- 
ably adds to their effectiveness. Blasting under water has been 
effectively accomplished by merely pouring nitro-glycerine into 
the drilled holes through a tube and then exploding the charge 
without any taniping except that furnished by the superincum- 
bent water. It has been found that air-spaces about a charge 
make a material reduction in the effectiveness of the explosion. 
It is therefore necessary to carefully ram the explosive into a 
solid mass. Of course the liquid nitro-glycerine needs no ram- 
ming, but dynamite should be rammed with a wooden rammer. 
Iron should be carefully avoided in ramming gunpowder. A 
copper bar is generally used. 

122. Exploding the charge. Black powder is generally ex- 
ploded by means of a fuse which is essentially a cord in which 
there is a thin vein of gunpowder, the cord being protected by 
tar, extra linings of hemp, cotton, or even gutta-percha. The 
fuse is inserted into the middle of the charge, and the tamping 
carefully packed around it so that it will not be injured. To 
produce the detonation required to explode nitro-glycerine and 
dynamite, there must be an initial explosion of some easily 
ignited explosive. This is generally accomplished by means of 
eaps containing fulminating-powder wdiicli are exploded by 
electricity. The electricity (in one class of caps) heats a very 
fine platinum wire to redness, thereby igniting the sensitive 
powder, or (in another class) a spark is caused to jump through 
the powder between the ends of two wires suitably separated. 
DyKamite can also be exploded by using a small cartridge of 
gunpowder which is itself exploded by an ordinary fuse. 

123. Cost. Trautwine estimates the cost of blasting (for 
mere excavation) as averaging 45 cents per cubic yard, falling 
as low as 30 cents for easy but hrittle rock, and running up to 



148 BAILROAD CONSTRUCTION. § 124. 

60 cents and even $i when the cutting is shallow, the rock 
especially tough, and the strata unfavorably placed. Soft tough 
rock frequently requires more powder than harder brittle 
rock. 

124. Classification of excavated material. The classification of 
excavated material is a fruitful source of dispute between con- 
tractors and railroad companies, owing mainly to the fact that 
the variation between the softest earth and the hardest rock is 
so gradual that it is very difficult to describe distinctions between 
different classifications which are unmistakable and indisputable. 
The classification frequently used is {a) earth, (J)) loose rock, and 
(c) solid rock. As blasting is frequently used to loosen "loose 
rock " and even "earth " (if it is frozen), the fact that blasting 
is employed cannot be used as a criterion, especially as this 
would (if allowed) lead to unnecessary blasting for the sake of 
classifying material as rock. 

Earth. This includes clay, sand, gravel, loam, decomposed 
rock and slate, boulders or loose stones not greater than 1 cubic 
foot (3 cubic feet, P. R. R.), and sometimes even "hard-pan." 
In general it will signify material which can be loosened by a 
plough with two horses, or with which one picker can keep one 
shoveller busy. 

Loose rock. This includes boulders and loose stones of more 
than one cubic foot and less than one cubic yard ; stratified rock, 
not more than six inches thick, separated by a stratum of clay ; 
also all material (not classified as earth) which may be loosened 
by pick or bar and which '^ can be quarried without blasting, 
although blasting may occasionally be resorted to. ' ' 

Solid rock includes all rock found in masses of over one cubic 
yard which cannot be removed except by blasting. 

It is generally specified that the engineer of the railroad 
company shall be the judge of the classification of the material, 
but frequently an appeal is taken from his decisions to the courts. 

125. Specifications for earthwork. The following specifica- 
tions, issued by the Norfolk and Western R. R., represent the 
average requirements. It should be remembered that very strict 



§ 125. EARTHWORK. 149 

specifications invariably increase the cost of the work, and fre- 
quently add to the cost more than is gained by improved quality 
of work. 

1. The grading will be estimated and paid for by the cubic 
yard, and will include clearing and grubbing, and all open ex- 
cavations, channels, and embankments required for the forma- 
tion of the roadbed, and for turnouts and sidings; cutting all 
ditches or drains about or contiguous to the road; digging the 
foundation-pits of all culverts, bridges, or walls ; reconstructing 
turnpikes or common roads in cases where they are destroyed or 
interfered with ; changing the course or channel of streams ; and 
all other excavations or embankments connected with or incident 
to the construction of said Railroad. 

2. All grading, except where otherwise specified, whether 
for cuts or fills, will be measured in the excavations and will be 
classified under the following heads, viz. : Solid Rock, Loose 
Rock, Hard-pan, and Earth. 

Solid Rock shall include all rock occurring in masses Avhich, 
in the judgment of the said Engineer Maintenance of AVay, may 
be best removed by blasting. 

Loose Rock shall include all kinds of shale, soapstone, and 
other rock Avhich, in the judgment of the said Engineer Main- 
tenance of Way, can be removed by pick and bar, and is soft 
and loose enough to be removed without blasting, although 
blasting may be occasionally resorted to ; also, detached stone of 
less than one (1) cubic yard and more than one (1) cubic foot. 

Hard-pan shall consist of tough indurated clay or cemented 
gravel, which requires blasting or other equally expensive 
means for its removal, or which cannot be ploughed with less 
than four horses and a railroad plough, or which requires two 
pickers to a shoveller, the said Engineer Maintenance of Way 
to be the judge of these conditions. 

Earth shall include all material of an earthy nature, of 
whatever name or character, not unquestionably loose rock or 
hard-pan as above defined. 

Powder. The use of powder in cuts will not be considered 



150 RAILROAD CONSTRVCTION, § 125 

iis a reason for any other classification than earth, unless the 
material in the cut is clearly other than earth under the above 
specifications. 

3. Earth, o^ravel, and other materials taken from the exca- 
vations, except when otherwise directed by the said Engineer 
Maintenance of Way or his assistant, shall be deposited in the 
adjacent embankment; the cost of removing and depositing 
which, when the. distance necessary to be hauled is not niore 
than sixteen hundred (1600) feet, shall be included in the price 
paid for the excavation. 

4. ExTEA Haul will be estimated and paid for as follows : 
w^henever material from excavations is necessarily hauled a 
greater distance than sixteen hundred (1600) feet, there shall be 
paid in addition to the price of excavation the price of extra 
haul per 100 feet, or part thereof, after the first 1600 feet; the 
necessary haul to be determined in each case by the said Engi- 
neer Maintenance of AVay or his assistant, from tlie profile and 
<3ross- sections, and the estimates to be in accordance therewith. 

5. All embankments shall be made in layers of such thick- 
ness and carried on in such manner as the said Eno^ineer Mainte- 
nance of Way or his assistant may prescribe, the stone and 
heavy materials being placed in slopes and top. And in com- 
pleting the fills to the proper grade such additional heights and 
fulness of slope shall be given them, to provide for their settle- 
ment, as the said Engineer Maintenance of Way, or his assistant, 
may direct. Embankments about masonry shall be built at 
such times and in such manner and of such materials as the said 
Engineer Maintenance of Way or his assistant may direct. 

6. In procuring materials for embankments from without 
the line of the road, and in wasting materials from cuttings, the 
place and manner of doing it shall in each case be indicated by 
the Engineer Maintenance of Way or his assistant; and care 
must be taken to injure or disfigure the land as little as possible. 
Borrow-pits and spoil-banks must be left by the Contractor in 
regular and sightly shape. 

7. The lands of the said Railroad Company shall be cleared 



§ 125. EARTHWORK. lol 

to the extent required by the said Engineer ^laintenance of 
AVay, or his assistant, of all trees, Inrushes, logs, and other 
perishable materials, which shall be destroyed by burning or 
deposited in heaps as the said Engineer Maintenance of Way, 
or his assistant, may direct. Lai'ge trees must be cut not more 
than two and one-half (2 J) feet from the ground, and under 
embankments less than four (4) feet high they shall be cut close 
to the o:round. All small trees and bushes shall be cut close to 
the ground. 

8. Clearing shall be estimated and paid for by the acre or 
fraction of an acre. 

9. All stumps, roots, logs, and other obstructions shall be 
grubbed out, and removed from all places where embankments 
occur less than two (2) feet in height ; also, from all places 
where excavations occur and from such other places as the said 
Engineer Maintenance of Way or his assistant may direct. 

10. Grnbbing shall be estimated and j^aid for by the acre or 
fraction of an acre. 

11. Contractors, when directed by the said Engineer Main- 
tenance of Way or his assistant in charge of the work, will 
deposit on the side of the road, or at such convenient points as 
may be designated, any stone, rock, or other materials that they 
may excavate; and all materials excavated and deposited as 
above, together with all timber removed from the line of the 
road, will be considered the property of the Kailroad Company, 
and the Contractors upon the respective sections will be respon- 
sible for its safe-keeping until removed by said Railroad Com- 
pany, or nntil their work is finished. 

12. Contractors will be accountable for the maintenance 
of safe and convenient places wherever public or private roads 
are in any way interfered with by them during the progi*ess of 
the work. They will also be responsible for fences thrown 
down, and for gates and bars left open, and for all damages 
occasioned thereby. 

18. Temporary bridges and trestles, erected to facilitate the 
progress of the work, in case of delays at masonry structures 



152 RAILROAD CONSTRUCTION. § 125. 

from any cause, or for other reasons, will be at the expense of 
the Contractor. 

Itt. The line of road or the gradients niaj be changed in. 
any manner, and at any time, if tlie said Engineer Maintenance 
of Way or his assistant shall consider such a change necessary 
or expedient ; but no claim for an increase in prices of excava- 
tion or embankment on the part of the Contractor will be allowed 
or considered unless made in writing before the work on that 
part of the section where the alteration has been made shall have 
been commenced. The said Engineer Maintenance of Way or 
his assistant may also, on the conditions last recited, increase or 
diminish the length of any section for the purpose of more 
nearly equalizing or balancing the excavations and embankments, 
or for any other reason. 

15. The roadbed will be graded as directed by the said En- 
gineer Maintenance of Way or his assistant, and in conformity 
with such breadths, depths, and slopes of cutting and filling as 
he may prescribe from time to time, and no part of the work 
will be finally accepted until it is properly completed and dressed 
o£E at the required grade. 



CHAPTER lY. 

TRESTLES. 

126. Extent of use. Trestles constitute from 1 to Sfo of the 
length of the average raih-oad. It was esthnated in 1889 that 
there was then about 2400 miles of single-track railway trestle 
in the United States, divided among 150,000 structures and 
estimated to cost about $75,000,000. The annual charire for 
maintenance, estimated at \ of the cost, therefore amounted to 
about $9,500,000 and necessitated the annual use of perhaps 
300,000,000 ft. B.M. of timber. The corresponding figures at 
the present time must be somewhat in excess of this. The 
magnitude of this use, which is causing the rapid disappearance 
of forests, has resulted in endeavors to limit the use of timber 
for this purpose. Trestles may be considered as justifiable under 
the following conditions : 

a. Permanent trestles. 

1. Those of extreme height — then called viaducts and fre- 
quently constructed of iron or steel, as the Kinzua viaduct, 302 
ft. high. 

2. Those across waterways — e.g.^ that across Lake Pontchar- 
train, near IS'ew Orleans, 22 miles long. 

3. Those across swamps of soft deep mud, or across a river- 
bottom, liable to occasional overflow. 

h. Temporary trestles. 

1. To open the road for traffic as quickly as possible — often 
a reason of great financial importance. 

2. To quickly replace a more elaborate structure, destroyed 

153 



154 RAILROAD CONSTRUCTION. § 127. 

by accident, on a road already in operation, so that the inter- 
ruption to traffic shall be a minimum. 

3. To form an earth embankment with earth brought from 
a distant point by the train-load, when such a measure would 
cost less than to borrow earth in the immediate neighborhood. 

4. To bridge an opening temporarily and thus allow time to 
learn the regimen of a stream in order to better proportion the 
size of the waterway and also to facilitate bringing suitable stone 
for masonry from a distance. In a new country there is always 
the double danger of eitlier bnilding a culvert too small, requir- 
ing expensive reconstruction, perhaps after a disastrous washout, 
or else wasting money by constructing the culvei't unnecessarily 
large. Much masonry has been built of a very poor quality of 
stone because it could be conveniently obtained and because good 
stone was unobtainable except at a prohibitive cost for transpor- 
tation. Opening the road for traffic by the nse of temporary 
trestles obviates both of these difficulties. 

127. Trestles vs. embankments. Low embankments are very 
much cheaper than low trestles both in first cost and mainte- 
nance. Yery high embankments are yqyj expensive to construct, 
but cost comparatively little to maintain. A trestle of equal 
height may cost much less to construct, but will be expensive to 
maintain — perhaps -J of its cost per year. To determine the 
height beyond which it will be cheaper to maintain a trestle 
rather than build an embankment, it will be necessary to allow 
for the cost of maintenance. The height will also depend on 
the relative cost of timber, labor, and earthwork. At the pres- 
ent average values, it will be found that for less heights than 
25 feet i\\e first cost of an embankment will generally be less 
than that of a trestle; this implies that a permanent trestle 
should never be constructed with a height less than 25 feet 
except for the reasons given in § 126. The height at which a 
permanent trestle is certainly cheaper than earthwork is more 
uncertain. A high grade line joining two hills will invariably 
imply at least a culvert if an embankment is used. If the 
culvert is built of masonry, the cost of the embankment will be 



§ 129. TRESTLES. 155 

so increased that the lieiglit at wliicli a trestle becomes economi- 
cal will be materially reduced. The cost of an embankment 
increases much more rai^idlv than the heiu'lit — with very liitrh 
embankments more nearly as the square of the height — while 
the cost of trestles does not increase as rapidly as the height. 
Although local circumstances may modify the application of any 
set rules, it is probably seldom that it will be cheaper to build 
an embankment 40 or 50 feet high than to permanently maintain 
a wooden trestle of that height. A steel viaduct would proba- 
bly be the best, solution of such a case. These are frequently 
used for permanent structures, especially when very high. The 
cost of maintenance is nuich less than that of wood, which 
makes the use of iron or steel preferable for permanent trestles 
unless wood is abnormally cheap. Neither the cost nor the con- 
struction of iron or steel trestles will be considered in this chapter. 

128. Two principal types. There are two- principal types of 
wooden trestles — pile trestles and framed trestles. The great 
objection to pile trestles is the rapid rotting of the portion of 
the pile which is underground, and the difficulty of renewal. 
The maximum height of pile trestles is about 30 feet, and even this 
height is seldom reached. Framed trestles have been constructed 
to a height of considerably over 100 feet. They are frequently 
built in such a manner that any injured piece may be readily 
taken out and renewed without interfering with traffic. Trestles 
consist of two parts — the supports called '* bents," and the 
stringers and floor system. As the stringers and floor system 
are the same for both pile and framed trestles, the " bents" are 
all that need be considered separately. 

PILE TRESTLES. 

129. Pile bents. A pile bent consists generally of four })iles 
driven into the ground deep enough to afford not only sufficient 
vertical resistance but also lateral resistance. On top of these 
piles is placed a horizontal ''cap." The caps are fastened to 
the tops of the piles by methods illustrated in Fig. 66. The 



156 



RAILROAD CONSTRUCTION. 



§129. 



method of fastening shown in each case should not be considered 
as applicable only to the particular tyj)e of j)ile bent used to illus- 
trate it. Fig. ^^ {a and d) illustrates a mortise- joint with a hard- 




FiG. 66. 

wood pin about 1^" in diameter. The hole for the pin should 
be bored separately tlirough the cap and the mortise, and 
the hole through the cap should be at a slightly higher 
level than that through the mortise, so that the cap will be 
drawn down tight when the j)in is driven. Occasionally an 
iron dowel (an iron pin about 1|" in diameter and about 6" 
long) is inserted partly in the cap and partly in the pile. The 
use of drift-bolts, shown in Fig. Q^ (h), is cheaj^er in first cost, but 
renders repairs and renewals very troublesome and expensive. 
'' Split caps," shown in Fig. 66 (r), are formed by bolting two 
half-size strips on each side of a tenon on top of the pile. 
Repairs are very easily and cheaply made without interference 
with the ti-affic and without injuring other pieces of the bent. 
The smaller pieces are more easily obtainable in a sound con- 
dition ; the decay of one does not affect the other, and the first 
cost is but little if any greater than the method of using a single 
piece. For further discussion, see § 186. 

For very light trafific and for a height of about 5 feet three 
vertical piles will sufiice, as shown in Fig. 66 (a). Up to a height 
of 8 or 10 feet four piles may be used without eway-bracino;, as 
in Fig. 66 (5), if the piles have a good bearing. For heights 
greater than 10 feet sway-bracing is generally necessary. The 
outside piles are frequently driven with a batter varying from 
1 : 12 to 1 : 4. 



§ 180. TRESTLES. 157 

Piles are made, if possible, from timber obtained in the 
vicinity of the work.- Durability is tlie great requisite rather 
than strength, for almost any timber is strong enough (except 
as noted below) and will be suitable if it will resist rapid decay. 
The following list is quoted as being in the order of preference 
on account of durability : 



1. Red cedar 

2. Red cypress 

3. Pitch-pine 

4. Yellow pine 



5. Wliile piue 

6. Redwood 

7. Elm 

8. Spruce 



9. White ouk 

10. Post-oak 

11. Red oak 



12. Black oak 

13. Hemlock 

14. Tamarac 



Red-cedar piles are said to have an average life of 27 years 
with a possil)le. maximum of 50 years, but the timber is rather 
weak, and if exposed in a river to flowing ice or driftwood is 
apt to be injured. Under these circumstances oak is prefer- 
able, although its life may be only 13 to 18 years. 

130. Methods of driving piles. The following are the prin- 
cipal methods of driving piles : 

a. A hammer weighing 2000 to 3000 lbs. or more, sliding 
in guides, is drawn up by horse- power or a portable engine, and 
allowed to id^W. freely. 

h. The same as above except that the hammer does not fall 
freely, but drags the rope and revolving drum as it falls and is 
thus quite materially retarded. The mechanism is a little more 
simple, but is less effective, and is sometimes made deliberately 
deceptive by a contractor by retarding the blow, in order to 
apparently indicate the requisite resistance on the part of 
the pile. 

The above methods have tlie advantage that the mechanism 
is cheap and can be transported into a new country with com- 
parative ease, but the work done is somewhat ineffective and 
costly compared with some of the more elaborate methods 
given below. 

c. Gunjpowder pile-drivers^ which automatically explode a 
cartridge every time the hammer falls. The explosion not only 
forces the pile down, but throws up the hammer for the next 
blow. For a given height of fall the effect is therefore doubled. 
It has been shown by experience, however, tliat when it is at- 



158 RAILROAD CONSTRUCTION. § 130, 

tempted to use such a pile-driver rapidly the meclianism be- 
comes so heated that the cartridges explode prematurely, and the 
method has therefore been abandoned. 

d. Steam pile-drivers, in which the hammer is operated 
directly by steam.' The hammer falls freely a height of about 
40 inches and is raised again by steam. The effectiveness is 
largely due to the rapidity of the blows, which does not allow 
time between the blows for the ground to settle around the pile 
and increase the resistance, which does happen w^hen the blows 
are infrequent. "The hammer-cylinder weighs 5500 lbs., and 
with 60 to 75 lbs. of steam gives 75 to 80 blows per minute. 
With 11 blows a large unpointed pile was driven 35 feet into a 
hard clay bottom in half a minute." Such a driver would cost 
about $800. 

The above four methods are those usual for dry earth. 
In very soft wet or sandy soils, where an unlimited supply of 
water is available, the water-jet is sometimes employed. A pipe 
is fastened along the side of the pile and extends to the pile- 
point. If water is forced through the pipe, it loosens the sand 
around the point and, rising along the sides, decreases the side 
resistance so that the pile sinks by its own weight, aided perhaps 
by extra weights loaded on. This loading may be accomplished 
by connecting the top of the pile and the pile-driver by a block 
and tackle so that a portion of the weight of the pile-driver is 
continually thrown on the pile. 

Excessive driving frequently fractures the pile below the 

surface and thereby greatly weakens its bearing power. To 

prevent excessive "brooming" of \\\q top of the 

pile, owing to the action of the hammer, the top 

should be protected by an iron ring fitted to the top 

of the pile. The "brooming" not only renders the 

driving ineffective and hence uneconomical, but 

vitiates the value of any test of the bearing power 

of the pile by noting the sinking due to a given 

Fig. 67. weight falling a given distance. If the ])ile is so 

soft that brooming is unavoidable, the top should be adzed of[ 




§ 131. TRESTLES. I59 

frequently, and especially should it be done just before the linal 
blows which are to test its bearing-power. 

In a new country judgment and experience will be required 
to decide intelligently whether to employ a simple drop-hanmier 
machine, operated by horse-power and easily transported but 
uneconomical in oi)eration, or a more complicated machine 
working cheaply and effectively after being transported at 
greater expense. 

131. Pile-driving formulae. If R = the resistance of a pile, 
and s the set of the pile during the last blow, iv the weight of 
the pile-hammer, and h the fall during the last blow, tl^en we 

may state the approximate relation that 7?^ = w/i, or A* = — 

s 
This is the basic principle of all rational formulae, but the 
maximum weight which a pile will sustain after it has been 
driven some time is by no means equal to the resistance of the 
pile during the last blow. There are also many other modi- 
fying elements which have been variously allowed for in the 
many proposed formulre. The formuh^ range from the extreme 
of empirical simplicity to very complicated attempts to allow 
properly for all modifying causes. As the simplest rule, 
specifications sometimes require that the piles shall be driven 
until the pile will not sink more than 5 inches under five 
consecutive blows of a 2000 11)., hammer falling 25 feet. 
The ''Engineering News formula"-^ gives the safe load as 
2w;A 
^-^pj, m which i^ = weight of hammer, /? = fall \\\ feet, 

s = set of pile in inc/tes under the last blow. This formula is 
derived from the above basic formula by calling the safe load | 
of the final resistance, and by adding (arbitrarily) 1 to the final 
set (s) as a conq^ensation for the extra resistance caused by the 
setthng of earth around the pile between each blow. This 
formula is used only for ordinary hammer-driving. When the 
piles are driven by a steam pile-driver the formula becomes 

* Engineering Neics, Nov. IT, 1892. 



160 



RAILROAD CONSTRUCTION. 



§132. 



safe load = 



2iuh 



Foi tlie '' gunpowder pile-driver," since 



5 + 0.1* 

the explosion of the cartridge drives the pile in with the same 
force with w^hich it throws the hammer upward, the effect is 
tivice that of the fall of the hammer, and the formula becomes 

safe load = — ; — ;— --. In these last two formulae the constant 

s -\- 0.1 

in the denominator is changed from .9 + 1 to 5+0.1. The 
constant (1.0 or 0.1) is supposed to allow, as before stated, for the 
effect of the extra resistance caused by the earth settling around the 
pile between each blow. The more rapid the blows tlie less the 
opportunity to settle and the less the proper value of the constant. 
The above formulae have been given on account of their 
simplicity and their practical agreement with experience. Many 
other formulae have been j)roposed, the majority of which are 
more complicated and attempt to take into account the weight of 
the pile, resistance of the guides, etc. While these elements, 
as well as many others, have their influence, their effect is so 
overshadowed by the indeterminable effect of other elements — as, 
for example, the effect of the settlement of earth around the pile 
between blows — that it is useless to attempt to employ anything 
but a purely empirical formula. 

132. Pile-points and pile-shoes. Piles are generally sharpened 
to a blunt point. If the pile is liable to strike boulders, sunken 
logs, or other obstructions which are liable to turn the point, it 

is necessary to protect the point by some 
form of shoe. Several forms in cast iron 
have been used, also a wrought-iron shoe, 
having four ' ' straps ' ' radiating from the 
apex, the straps being nailed on to the pile, 
as shown in Fig. %S (h). The cast-iron 
form show^n in Fig. 68 {a) has a base cast 
around a drift-bolt. The recess on the top 
of the base receives the bottom of the pile 
and prevents a tendency to split the bottom 
of the pile or to force the shoe off laterally. 




Ficx. 68. 



§ 134. TRESTLES. 161 

133. Details of design. Xo theoretical calculations of the 
strength of pile bents need be attempted on account of the ex- 
treme complication of the theoretical strains, the uncertainty as 
to the real strength of the timber used, the variability of that 
strength with time, and the insigniiicance of the economy that 
would be possible even if exact sizes could be computed. The 
piles are generally required to be not less than 10" or 12" in 
diameter at the large end. The P. E. E. requires that they shall 
be "not less than 14 and 7 inches in diameter at butt and small 
end respectively, exclusive of bark, which must be removed." 
The removal of the bark is generally required in good work. 
Soft durahle woods, such as are mentioned in § 129, are best 
for the piles, but the caps are generally made of oak or yellow 
pine. The caps are generally 14 feet long (for single track) 
with a cross-section 12'' X 12'' or 12" X 14". "Split caps" 
would.consist of two pieces Q" X 12". The sway-braces, never 
used for less heights than 6', are made of 3" X 12" timber, and 
are spiked on with f " spikes 8" long. The floor system will be 
the same as that descri])ed later for framed trestles. 

134. Cost of pile trestles. The cost, per linear foot, of piling 
depends on the method of driving, the scarcity of suitable tim- 
ber, the price of labor, the length of the piles, and the amount 
of shifting of the pile-driver required. The cost of soft-wood 
piles varies from 8 to 15 c. per lineal foot, and the cost of oak 
piles varies from 10 to 30 c. per foot according to the length, 
the longer piles costing more per foot. The cost of driving will 
average about $2.50 per j^ile, or 7.5 to 10 c. per lineal foot. 
Since the cost of shifting the pile-driver is quite an item in tlie 
total cost, the cost of driving a long pile would be less per foot 
than for a short pile, but on the other hand the cost of the pile' 
is greate^^ per foot, which tends to make the total cost per 
foot constant. Specifications generally say that the piling will 
be paid for per lineal foot of piling left in the worl. The wast- 
age of the tops of piles sawed off is always something, and is 
frequently very large. Sometimes a small amount per foot of 
piling sawed off is allowed the contractor as compensation for 



162 



BAILROAD CONSTRUCTION. 



§135. 



liis loss. This reduces the contractor's risk and possibly reduces 
his bid by an equal or greater amount than the extra amount 
actually paid him. 



FRAMED TRESTLES. 



135. Typical Design. A typical design for a framed trestle 
bent is given in Fig. 69. This represents, with slight variations 
of detail, the plan according to which a large part of the framed 




Fig. 69. 

trestle bents of the country have been built — i.e., of those less 
than 20 or 30 feet in height, not requiring multiple-story 
construction. 

136. Joints, (a) The mortise-and-tenon joint is illustrated in 
Fig. 69 and also in Fig. %Q (a). The tenon should be about 
3" thick, 8'' wide, and 5^" long. The mortise should be cut 
a little deeper than the tenon. "Drip-holes " from the mortise 
to the outside will assist in draining off water that 
may accumulate in the joint and thus prevent the 
rapid decay that would otherwise ensue. These 
joints are very troublesome if a single post decays 
and requires renewal. It is generally required that 
Fig. 70. the mortise and tenon should be thoroughly daubed 

with paint before putting them together. This will tend to 



r\M 



m 



m 



HOLE 



§ 137. 



TRESTLES. 



163 




make the joint watei'-tiglit and prevent decay from the ac- 
cumulation and retention of water in the joint. 

(b) The plaster joint. This joint is made by bolting and 
spiking a 3" X 12" plank on both 
sides of the joint. The cap and 
sill should be notched to receive 
the posts. Repairs are greatly 
facilitated by the use of these 
joints. This method has been 
used by the Delaware and Hud- 
son Canal Co. [R. R.]. 

(c) Iron plates. An iron plate of the form shown in Fig. 72 
(b) is bent and used as shown in Fig. 72 {a). Bolts passing through 

the bolt-holes shown secure the 
plates to the timbers and make a 
strong joint which may be readily 
loosened for repairs. By slight 
c modifications in the design the 
method may be used for inclined 



Fig. 71. 





.:::i:/i 


^^^1 


a 






Pt'-' 


^ ^ 


^'i> 


--■'(a) 



-J 



(&) 



c posts and complicated joints. 



Ftg 72. 



(d) Split caps and sills. These 
are described in ^ 129. Their 
advantages apply with even greater "force to framed trestles. 

(e) Dowels and drift-bolts. These joints facilitate cheap and 
rapid construction, but renewals and repairs are very difficult, it 
being almost impossible to extract a drift-bolt which has been 
driven its full length without splitting open the pieces contain- 
ing it. Notwithstanding this objection they are extensively 
used, especially for temporary work which is not expected to 
be used long enough to need repairs. 

137. Multiple-story construction. Single-story framed trestle 
bents are used for heights up to 18 or 20 feet and excejitionally 
up to 30 feet. For greater heights some such construction 
as is illustrated in a skeleton design in Fig. 73 is used. By 
using split sills between each story and separate vertical and 
batter posts in each story, any piece may readily be removed and 



164 



RAILROAD CONSTRUCTION. 



138. 



renewed if necessary. The height of these stories varies, in 

different designs, from 15 to 25 and 
even 30 feet. In some designs tlie 
structure of each storj is independent 
of the stories above and below. This 
greatly facilitates both the original con- 
struction and subsequent repairs. In 
other designs the verticals and batter- 
posts are made continuous through two 
consecutive stories. The structure is 
somewhat stiffer, but is much more diffi- 
cult to repair. 

Since the bents of any trestle are 

usually of variable height and those 

Fig. 73. heights are not always an even multiple 

of the uniform height desired for the stories, it becomes 

necessary to make the upper stories of unifoi-m height and let 





Fig. 74. 



the odd amount go to the lowest story, as shown in Figs. 73 
and 74. 

138. Span. The shorter the span the greater the number 
of trestle bents; the longer the span the greater the required 
strength of the stringers supporting the floor. Economy de- 
mands the adoption of a span that shall make the sum of these 
requirements a minimum. The liigher the trestle the greater 
the cost of each bent, and the greater the span that would be 
justifiable. Nearly all trestles have bents of variable height, 
but the advantage of employing uniform standard sizes is so 
great that many roads use the same span and sizes of timber not 
only for the panels of any given trestle, but also for all trestles 



§139 



TRESTLES. 



165 



regardless of height. The spans generally used vary from 10 
to 16 feet. The Norfolk and Western E. K. uses a span of 
12' Q" for all single-story trestles, and a span of 25' for 
all multiple-story trestles. The stringers are the same in both 
cases, but wheu the span is 25 feet, knee- braces are run 




Fig. 75. 

from the sill of the lirst story below to near the middle of each 
set of stringers. These knee-braces are connected at the top by 
a ^'straining-beam" on which the stringers rest, thus support- 
ing the stringer in the center and virtually reducing the span 
about one-half. 

139. Foundations, (a) Piles. Piles are frequently used as a 
foundation, as in Fig. 76, particularly in soft ground, and also 
for temporary structures. These 
foundations are cheap, quickly con- 
structed, and are particularly valuable 
when it is financially necessary to open 



(^Va 



^fAJ 



SILL 



tne road tor tratlic as soon as possible ^^m^^M^m^mm^^ 
and with the least expenditure of ''"^ ''■"' U U 
money; but there is the disadvantage Fig. 76. 

of inevitable decay within a few years unless the piles are chemi- 
cally treated, as will be discussed later. Chemical treatment, 
however, increases the cost so that such a foundation would 
often cost more tlian a foundation of stone. A pile should be 
driven under each post as shown in Fig. 76. 



166 



RAILROAD CONSTRUCTION. 



§140. 



(b) Mud-sills. Fig. 77 

n w/ n 



illustrates the use of mud-sills as 
built bj the Louisville and Nash- 
ville E. K. Eight blocks 12" X 12" 
X 6' are used under each bent. 
When the ground is very soft, two 
additional timbers (12" X 12" X 
length of bent- sill), as shown by the 
dotted lines, are placed underneath. 
Fig. 77. The number required evidently de- 

pends on the nature of the ground. 

(c) Stone foundations. Stone foundations are the best and 
the most expensive. For very high trestles the JS'orfolk and 
Western R.R. employs foundations as shown in Fig. 78, the 

* SILL 0!^ TRESTLE 



r 

L - 


-- 


-- 





-- 


-- 


-- 




-- 


:: 


- -i 
-J 


1 SILL 1 


i- 


-- 


- 




-- 


-- 


-- 





-- 


-- 


-J 






< ^13 > " — 8 — ^ , -^ 13 > 

Fig. 78. 

walls being 4 feet thick. When the height of the trestle is 72 
feet or less (the plans requiring for 72' in height a foundation- 
wall 39' 6' long) the foundation is made continuous. The sill 
of the trestle should rest on several short lengths of 3" X 12'' 
plank, laid transverse to the sill on top of the wall. 

140. Longitudinal bracing. This is required to give the 
structure longitudinal stiifness and also to reduce the columnar 
length of the posts. This bracing generally consists of hori- 
zontal " waling-strips " and diagonal braces. Sometimes the 
braces are placed wholly on the outside posts unless the trestle 
is very high. For single-story trestles the P. R. R. employs 
the "laced "' system, i.e., a line of posts joining the cap of one 
bent with the sill of the next, and the sill of that bent with the 
cap of the next. Some plans employ braces forming an X in 
alternate panels. Connecting these braces in the center more 
than doubles their columnar strength. Diagonal braces, when 
bolted to posts, should be fastened to them as near the ends of 



143. 



TI^ESTLES. 



1G7 



the posts as possible. The sizes employed vary largely, depend- 
ing on the clear length and on whether they are expected to act 
by tension or compression. 3" X 12" planks are often used 
when the design would require tensile strength only, and 
S" X S" posts are often used when compression may be 
expected. 

141. Lateral bracing. Several of the more recent designs of 
trestles employ diagonal lateral bracing between the caps of 
adjacent bents. It adds greatly to the stiffness of the trestle 
and better maintains its alignment. 6" X 6" posts, formincr 
an X and connected at the center, will answer the purpose. 

142. Abutments. When suitable stone for masonry is at 
hand and a suitable subsoil for a foundation is obtainable without 
too much excavation, a masonry abutment will be the best. 
Such an abutment would probably be used when masonry 
footings for trestle bents were employed (§ 139, c). 

Another method is to construct a "crib" of 10" x 12" 
timber, laid horizontally, drift-bolted together, securely braced 
and embedded into the ground. Except for temporary con- 
struction such a method is generally objectionable on account of 
rapid decay. 

Another method, used most commonly for pile trestles, and 
for framed trestles having pile 
foundations (§ 139, a\ is to use a 
pile bent at such a place that the 
natural surface on the up-hill side 
is not far below the cap, and the 
thrust of the material, filled in to 
bring the surface to grade, is insig- 
nificant. 3" X 12" planks are placed Fig. 79. 
behind the piles, cap, and stringers to retain the filled material. 




FLOOR SYSTEMS. 

143. stringers. The general practice is to use two, three, 
and even four stringers under each rail. Sometimes a strin<rer 



168 



RAILBOAD CONSTRUCTION. 



§144 



is placed under eacli guard-rail. Generally the stringers are 
made of two panel lengths and laid so that the joints alternate. 
A few roads use stringers of only one panel length, but this 
practice is strongly condemned by many engineers. The 
stringers should be separated to allow a circulation of air around 
them and prevent the decay which would occur if they were 
placed close together. This is sometimes done by means of 2'' 
planks, 6' to 8' long, which are placed over each trestle bent. 
Several bolts, passing through all the stringers forming a group 
and through the separators, bind them all into one solid con- 
struction. Cast-iron ''spools" or washers, varying from 4z" to 
^" in length (or thickness), are sometimes strung on each bolt so 
as to separate the stringers. Sometimes washers are used 
between the separating planks and the stringers, the object of 
the separating planks then being to bind the stringers, especially 
abutting stringers, and increase their stiffness. 

The most common size for stringers is 8'^ X 16''. The 
Pennsylvania Railroad varies the width, depth, and number of 
stringers under each rail according to the clear span. It may 
be noticed that, assuming a uniform load per running foot, both 



Clear span. 


No. of pieces 
under each rail. 


Width. 


Depth. 


10 feet 
12 " 
14 " 
16 " 


2 
2 
2 
3 


8 iuches 
8 '' 
10 " 
8 " 


15 inches 

16 " 

17 " 
17 " 



the pressure per square inch at the ends of the stringers (the 
caps having a width of 12") and also the stress due to trans- 
verse strain are kept approximately constant for the variable gros:* 
load on these varying spans. 

144. Corbels. A corbel (in trestle-work) is a stick of timber 
(perhaps two placed side by side), about 3' to 6' long, placed 
underneath and along the stringers and resting on the cap. 
There are strong prejudices for and against their use, and a 



S 145. 



TRESTLES. 



169 



corresponding diversity in practice. They are bolted to tlie 
striijgers and thus stilten the joint. They certainly reduce the 
objectionable crushing of the libers at each end of the stringer, 
but if the corbel is no wider than the stringers, as is generally 
the case, the area of pressure between the corbels and the cap is 




Fig. 80. 

no greater and the pressure per square inch on the cap is no less 
than the pressure on the cap if no corbels were used. If the 
corbels and cap are made of hard wood, as is recommended by 
some, the danger of crushing is lessened, but the extra cost and 
the frequent scarcity of hard wood, and also the extra cost and 
labor of using, corbels, may often neutralize the advantages 
obtained by their use. 

145. Guard-rails. These are frequently made of b" X 8'' 
stuff, notched 1" for each tie. The sizes vary up to S" X 8'^ 
and the depth of notch from f" to 1^" . They are generally 
bolted to every third or fourth tie. It is frequently specified 
that they shall be made of oak, white pine, or yellow pine. The 
joints are made over a tie, by halving each piece, as illustrated 
in Fig. 81. The joints on opposite sides of the trestle should be 




Ficx. 81. 



''staggered." Some roads fasten every tie to the guard-rail, 
using a bolt, a spike, or a lag-screw. 

Guard-rails were originally used with the idea of preventing 
the wheels of a derailed truck from running off the ends of the 
ties. But it has been found that an outer guard-rail alone (with- 
out an inner guard-rail) becomes an actual element of danger, 
since it lias frequently happened that a derailed wheel has cauglit 



170 RAILROAD CONSTRUCTION. § 147. 

on the outer guard-rail, thus causing the truck to slew around 
and so produce a dangerous accident. The true function of the 
•outside guard-rail is thus changed to that of a tie-spacer, which 
keeps the ties from spreading when a derailment occurs. The 
inside guard-rail generally consists of an ordinary steel rail 
spiked about 10 inches inside of the running rail. These inner 
guard-rails should be bent inward to a point in the center of the 
-track about 50 feet from the end of the bridge or trestle. If 
the inner guard-rails are placed with a clear space of 10 inches 
inside the running rail, the outer guard-rails should be at least 
Q' 10'^ a]3art. They are generally much farther apart than this. 

146. Ties on trestles. If a car is derailed on a bridge or 
trestle, the heavily loaded wheels are apt to force their way be- 
tween the ties by displacing them unless the ties are closely 
spaced and fastened. The clear space between ties is generally 
equal to or less than their width. Occasionally it is a little more 
than their width. Q" X 8'^ ties, spaced 14:" to 16" from cen- 
ter to center, are most frequently used. The length varies from 
9' to 12' for single track. They are generally notched ^" deep 
on the under side where they rest on the stringers. Oak ties 
are generally required even when cheaper ties are used on the 
other sections of the road. Usually every third or fourth tie is 
bolted to the stringers. When stringers are placed underneath 
the guard-rails, bolts are run from the top of the guard-rail to 
the under side of the stringer. The guard-rails thus hold down 
the whole system of ties, and no direct fastening of the ties to 
the stringers is needed. 

147. Superelevation of the outer rail on curves. The location 
of curves on trestles should be avoided if possible, especially 
when the trestle is high. Serious additional strains are in- 
troduced especially when the curvature is sharp or the 
speed high. Since such curves are sometimes practically un- 
avoidable, it is necessary to design the trestle accordingly. 
If a train is stopped on a curved trestle, the action of the train on 
the trestle is evidently vertical. If the train is moving with a 
©onsiderable velocity, the resultant of the weight and the cen- 



§147. 



TRESTLES. 



171 



trifuixal action is a force somewhat inclined from tlie vertical. 
Both of these conditions may be expected to exist at times. If 
the axis of the system of posts is vertical (as illustrated in 
methods a, Z>, <?, d, and e), any lateral force, such as would be 
produced by a moving train, will tend to rack the trestle bent. 
If the stringers are set vertically, a centrifugal force likewise 
tends to tip them sidewise. If the axis of the system of posts 
(or of the stringers) is inclined so as to coincide w^ith the j^ressure 
of the train on the trestle when the train is moving at its normal 
velocity, there is no tendency to rack the trestle when the train 
is moving at that velocity, but there will be a tendency to rack 
the trestle or twist the stringers when the train is stationary. 
Since a moving train is usually the normal condition of affairs, 
as well as the condition which produces the maximum stress, an 
inclined axis is evidently preferable from a theoretical stand- 
point ; but whatever design is adopted, the trestle should evi- 
dently be sufficiently cross-braced for either a moving or a 
stationary load, and any proposed design must be studied as to 
the effect of hoth of these conditions. Some of the various 
methods of securing the requisite superelevation may be described 
as follows : 

(a) Framing the outer posts longer than the inner posts, so 
that the cap is inclined at the proper angle ; axis of posts verti- 
cal. (Fig. 82.) The method requires 
more work in framing the trestle, 
but simplifies subsequent track-laying 
and maintenance, unless it should be 
found that the superelevation adopted 
is unsuitable, in which case it could be 
corrected by one of the other methods 
given below. The stringers tend to 
twist when the train is stationary. Fia. 8? 

(b) Notching the cap so that the stringers are at a different 
elevation. (Fig. 83.) This weakens the cap and requires that 
all ties shall be notched to a bevelled surface to fit the striuircrs. 




172 



RAILROAD CONSTRUCTION. 



147. 




■which also weakens the ties. A centrifugal force will tend to 

twist the stringers and rack the trestle, 
(c) Placing wedges underneath the 
ties at each stringer. These wedges are 
fastened witli two bolts. Two or more 
wedges will be required for each tie. 
The additional number of pieces re- 
quired for a long curve wiJl be im- 
U mense, and the work of inspection and 
Fi<^ 83. keeping the nuts tight will greatly in- 

crease the cost of maintenance. 

(d) Placing a wedge under the outer rail at each tie. This 
requires but one extra piece per tie. There is no need of a 
wedge under the inner tie in order to make the rail normal to 
the tread. The resulting inward inchnatiori is substantially that 
produced by some forms of rail-chairs or tie-plates. The spikes 
(a little longer than usual) are driven through the wedge into 
the tie. Sometimes "lag-screws" are used instead of spikes. 
If experience proves that the superelevation is too much or too 
little, it may be changed by this method with less work than by 
any other. 

(e) Corbels of different heights. When corbels are used (see 
§ 144) the required inclination of the floor system may be ob- 
tained by varying the depth of the corbels. 

(f ) Tipping the whole trestle. 
This is done by placing the 
trestle on an inclined founda- 
tion. If very much inclined, 
the trestle bent must be secured 
against the possibility of slip- 
ping sidewise, for the slope 
would be considerable with a 
sharp curve, and the vibration ^ 
of a moving train would reduce Fig. 84. 
the coeflicient of friction to a comparatively small quantity. 

(g) Fra^ming the outer posts longer. This case is identical 




§ 149. TRESTLES. 173 

with case {a) except that the axis of the system of posts is 
inchiied, as in case (/), but the sill is horizontal. 

The above-described plans will suggest a great variety of 
methods which are possible and which dili'er from the above 
only in minor details. 

148. Protection from fire. Trestles are peculiarly subject to 
iire, from passing locomotives, which may not only destroy the 
trestle, but perhaps cause a terrible disaster. This danger is 
sometimes reduced by placing a strip of galvanized iron along 
the top of each' set of stringers and also along the tops of the 
caps. Still greater protection was given on a long trestle on the 
Louisville and Nashville R. R. by making a solid flooring of 
timber, covered with a layer of ballast on which the ties and 
rails were laid as usual. 

Barrels of water should be provided and kept near all trestles, 
and on very long trestles barrels of water should be placed every 
two or three hundred feet along its length. A place for the bar- 
rels may be provided by using a few ties which have an extra 
length of about four feet, thus forming a small platform, which 
should be surrounded by a railing. The track-walkers should be 
held accountable for the maintenance of a supply of water in 
these barrels, renewals being frequently necessary on account of 
evaporation. Such platforms should also be provided as refuge- 
bays for track- walkers and trackmen working on the trestle. On 
very long trestles such a platform is sometimes provided with 
sufiicient capacity for a hand-car. 

149. Timber. Any strong durable timber may be used when 
the choice is limited, but oak, pine, or cypress are preferred 
when obtainable. When all of these are readily obtainable, 
the various parts of the trestle will be constructed of different 
kinds of wood — the stringers of long-leaf pine, the posts and 
braces of pine or red cypress, and the caps, sills, and corbels (if 
used) of white oak. The use of oak (or a similar hard wood) 
for caps, sills, and corbels is desirable because of its greater 
strength in resisting crushing across the grain, which is the 
critical test for these parts. There is no physiological basis to 



174 EAILROAD CONSTRUCTION, § 151. 

the objection, sometimes made, that different species of timber, 
in contact with each other, will rot quicker than if only one 
kind of timber is used. When a very extensive trestle is to be 
built at a place where suitable growing timber is at hand but 
there is no convenient sawmill, it will pay to transport a port- 
able sawmill and engine and cut up the timber as desired. 

150. Cost of framed timber trestles. The cost varies widely 
on account of the great variation in the cost of timber. When 
a railroad is first penetrating a new and undeveloped region, the 
cost of timber is frequently small, and when it is obtainable from 
the company's right-of-way the only expense is felling and 
sawing. The work per M., B. M., is small, considering that a 
single stick 12'' X 12'' X 25' contains 300 feet, B. M., and 
that sometimes a few hours' work, worth less than $1, will 
finish all the work required on it. Smaller pieces will of course 
require more work per foot, B. M. Long-leaf pine can be pur- 
chased from the mills at from §8 to $12 per M. feet, B. M., 
according to the dimensions. To this must be added the freight 
and labor of erection. The cartage from the nearest railroad to 
the trestle may often be a considerable item. Wrought iron 
will cost about 3 c. per pound and cast iron 2 c, although the 
prices are often lower than these. The amount of iron used 
depends on the detailed design, but, as an average, will amount 
to $1.50 to $2 per 1000 feet, B. M., of timber. Alarge part of 
the trestling of the country has been built at a contract price of 
about $30 per 1000 feet, B. M., erected. While the cost will 
frequently rise to $10 and even $50 when timber is scarce, it 
will drop to $13 (cost quoted) when timber is cheap. 

DESIGN OF WOODEN TRESTLES. 

161. Common practice. A great deal of trestling has been 
constructed without any rational design except that custom and 
experience have shown that certain sizes and designs are probahly 
safe. This method has resulted occasionally in failures but 
more frequently in a very large waste of timber. Many railroads 



§ 152. TRESTLES. 175 

employ a uniform size for all posts, caps, and sills, and a 
uniform size for stringers, all regardless of the height or span of 
the trestle. For repair work there are practical reasons favoring 
this. "To attempt to run a large lot of sizes would be more 
wasteful in the end than to maintain a few stock sizes only. 
Lumber can be bought more cheaply by giving a general order 
for 'the run of the mill for the season,' or 'a cargo lot,' 
specifying approximate percentages of standard stringer size, of 
12 X 12-inch stuff, 10 X 10-inch stuff, etc., and a liberal pro- 
portion of 3- or tt-inch plank, all lengths thrown in. The 12 x 12- 
inch stuff, etc., is ordered all lengths, from a certain specified 
length up. In case of a wreck, washout, burn-out, or sudden 
call for a trestle to be completed in a stated time, it is much 
more economical and practical to order a certain number of 
carloads of ' trestle stuff ' to the ground and there to select piece 
after piece as fast as needed, dependent only upon the length of 
stick required. When there is time to make the necessary sur- 
veys of the ground and calculations of strength, and to wait for a 
special bill of timber to be cut and delivered, the use of differ- 
ent sizes for posts in a structure would be warranted to a certain 
extent."* For new construction, when there is generally 
sufficient time to design and order the proper sizes, such waste- 
fulness is less excusable, and under any conditions it is both 
safer and more economical to prepare standard designs which 
can be made applicable to varying conditions and which will at 
the same time utilize as much of the strength of the timber as 
can be depended on. In the following sections will be given 
the elements of the preparation of such standard designs, which 
will utilize uniform sizes with as little waste of timber as possible. 
It is not to be understood that special designs should be made 
for each individual trestle. 

152. Required elements of strength. The stringers of 
trestles are subject to transverse strains, to crushing across the 
grain at the ends, and to shearing along the neutral axis. The 

* From "Economical Desif^nini; of Timber Trestle Bridges." 



176 RAILROAD CONSTRUCTION'. § 153. 

strength of tlie timber must therefore be computed for all these 
kinds of stress. Caps and sills will fail, if at all, by crushing 
across the grain ; although subject to other forms of stress, these 
could hardly cause failure in the sizes usually employed. There 
is an apparent exception to this : if piles are improperly driven 
and an uneven settlement subsequently occurs, it may have the 
effect of transferring practically all of the weight to two or three 
piles, while the cajy is subjected to a severe transverse strain 
which may cause its failure. Since such action is caused gener- 
ally by avoidable errors of construction it may be considered as 
abnormal, and since such a failure will generally occur by a 
gradual settlement, all danger may be avoided by reasonable 
care in inspection. Posts must be tested for their columnar 
strength. These parts form the bulk of the trestle and are the 
parts which can be definitely designed from known stresses. 
The stresses in the bracing are more indefinite, depending on 
indeterminate forces, since the inclined posts take up an un- 
known proportion of the lateral stresses, and the design of the 
bracing may be left to what experience has shown to be safe, 
without involving any large waste of timber. 

153. Strength of timber. Until recently tests of the strength 
of timber have generally been made by testing small, selected, 
well-seasoned sticks of " clear stuff," free from knots or imper- 
fections. Such tests would give results so much higher than 
the vaguely known strength of large unseasoned " commercial " 
timber that very large factors of safety were recommended — 
factors so large as to detract from any confidence in the whole 
theoretical design. Recently the U. S. Government has been 
making a thoroughly scientific test of the strength of full-size 
timber under various conditions as to seasoning, etc. The work 
has been so extensive and thorough as to render possible the 
economical designing of timber structures. 

One important result of the investigation is the determina- 
tion of the great influence of the moisture in the timber and 
the law of its effect on the strength. It has been also shown 
that timber soaked with water has substantially the same 



153. 



TRESTLES. 



177 



strength as green timber, even tliough tlic timber had once been 
thoroughly seasoned. Since trestles are exposed to the weather 
they should be designed on the basis of using green timber. 
It has been shown that the strength of green timber is very 
regularly about 55 to 60^ of the strength of timber in which 
the moisture is 12^ of the dry weight, l^fo being the proportion 
of moisture usually found in timber that is protected from the 
weather but not heated, as, e.g., the timber in a barn. Since 
the moduli of rupture have all been reduced to this standard of 
moisture (12^), if we take one-eighth of the ruj^ture values, it 
still allows a factor of safety of about fixQ^ even on green timber. 

Moduli of rupture for various timbers. [\2% moisture.] 

(Coudeused from U. S. Forestry Circular, No. 15.) 



No. 



9 

10 



11 
12 

13 
14 
15 
16 
19 
20 



21 
27 
28 
29 
30 



Species. 



Long-leaf pine.. . . 

Cuban " 

Short-leaf " 

Loblollv " . . . . 

White' " 

Red " .... 

Spruce " .. . . 

Bald cypress 

White cedar 

Douglas spruce, . . 

White oak 

Overcup " 

Post 
Cow 
Red 

Texan " 

Willow " 

Spanish " 

Sliagbark hickory 
Pignut 

White elm , 

Cedar " 

White ash , 



Weight 

per 

cubic 

foot. 



38 

39 
32 
33 
24 
31 
39 



29 
23 
32 



50 
46 
50 
46 
45 
46 
45 
46 



51 
56 
34 
46 
39 



Cross-bending. 



Ultimate 
Strength. 



12 600 

13 600 
10100 
11300 

7 900 

9100 

10000 



7900 

6 300 

7 900 



13100 
11300 
12 300 
11500 
11400 
13100 
10400 
12 000 



16 000 
18 700 
10 300 
13 500 
10 800 



Modulus of 
Elasticity. 



2 070 ono 

2 370 000 

1 680 000 

2 050 000 
1 390 000 
1 620 000 
1 640 000 



1 290 000 
910 000 

1 680 000 



2 090 000 

1 620 000 

2 080 000 
1 610 000 
1 970000 
1860 000 
17^0 000 
19'50000 



2 390 000 
2 730 000 
1540 000 
1700000 
1640000 



Crush- 
ing end 
wise. 



8000 
8700 
6500 
7400 
5400 
6700 
7300 



6000 
5200 
5700 



8500 
7800 
7100 
7400 
7300 
8100 
7200 
7700 



9500 
10900 
6500 
8000 
7200 



Crush- 


Shear- 


ing 


ing 


across 


along 


gram. 


gram. 


1180 


700 


1220 


700 


960 


700 


1150 


700 


700 


400 


1000 


500 


1200 


800 


800 


500 


700 


400 


800 


500 


2200 


1000 


1900 


1000 


3000 


1100 


1900 


900 


2300 


1100 


2000 


900 


1600 


900 


1800 


900 


2700 


1100 


8200 


1200 


1200 


800 


2100 


1800 


1900 


1100 



178 



RAILROAD CONSTRUCTION. 



§153. 



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§154. 



TRESTLES. 179 



On page 177 there are quoted the vahies taken from the U. S. 
Government reports on the strength of timber, the tests probably 
beino- the most thorough and reliable that were ever made. 

On page 178 are given the "average safe allowable work- 
in o- unit stresses in pounds per square inch," as recommended 
b^^the committee on '' Strength of Bridge and Trestle Timbers," 
the work being done under the auspices of the Association of 
Eailway Superintendents of Bridges and Buildings. The report 
was presented at their fifth annual convention, held in New 
Orleans, in October, 1895. 

154. Loading. As shown in § 138, the span of trestles is 
always small, is generally 14 feet, and is never greater than IS' 
except when supported by knee-braces. The greatest load that 
will ever come on any one span will be the concentrated loading 
of the drivers of a consolidation locomotive. With spans of 14 
feet or less it is impossible for even the four pairs of drivers to 
be on the same span at once. The weight of the rails, ties, and 
guard-rails should be added to obtain the total load on the string- 
ers, and the weight of these, plus the weight of the stringers, 
should be added to obtain the pressure on the caps or corbels. 
This dead load is almost insignificant compared with the live 
load and may be included with it. The weight of rails, ties, 
etc., may be estimated at 200 pounds per foot To obtain 
the weight on the caps the weight of the stringers must be 
added, which depends on the design and on the weight per cubic 
foot of the wood employed. But as the weight of the stringers 
is comparatively small, a considerable percentage of variation 
in wei^'-ht will have but an insignificant effect on the result. 
Disreo-ardinfi- all refinements as to actual dimensions, the ordi- 
nary maximum loading for standard gauge railroads may be 
taken as that due to four pairs of driving-axles, spaced 5' 0" 
apart and giving a pressure of 25,000 pounds per axle. This 
should be increased to 40,000 pounds per axle (same spacing) 
for the heaviest trafiic. On the basis of 25,000 pounds per 
axle the following results have been computed : 



180 



RAILBOAD CONSTRUCTION'. 



155. 



STRESSES ON VARIOUS SPANS DUE TO MOVING LOADS OF 25,000 POUNDS, 

SPACED 5' 0" APART. 



Span in feet. 


Max. mom.— 
ft. lbs. 


Max. shear. 


Max load on 
one cap. 


10 
12 
14 
16 

18 


65 000 
103 600 
142 400 
181 400 
220 600 


38 500 
45 000 
49 600 
54 725 
60100 


52100 
62 700 
74 200 
85 700 
97 900 



Although the dead load does not vary in proportion to the 
live load, jet, considering the very small influence of the dead 
load, there will be no appreciable error in assuming the corre- 
sponding values, for a load of 40,000 lbs. per axle, to be |o ^f 
those given in the above tabulation. 

155. Factors of safety. — The most valuable result of the gov- 
ernment tests is the knowledge that under given moisture condi- 
tions the strength of various species of sound timber is not the 
variable uncertain quantity it was once supposed to be, but that 
its strength can be relied on to a comparatively close percentage. 
This confidence in values permits the employment of lower fac- 
tors of safety than have heretofore been permissible. Stresses, 
which when excessive would result in immediate destruction, 
such as cross- breaking and columnar stresses, should be allowed 
a higher factor of safety — say 6 or 8 for green timber. Other 
stresses, such as crushing across the grain and shearing along the 
neutral axis, which will be apparent to inspection before it is 
dangerous, may be allowed lower factors — say 3 to 5. 

156. Design of stringers. — The strength of rectangular beams 
of equal width varies as the square of the depth ; therefore deep 
beams are the strongest. On the other hand, when any cross- 
sectional dimension of timber much exceeds 12" the cost is 
much higher per M., B.M., audit is correspondingly difi^cult to 
obtain thoroughly sound sticks, free from wind-shakes, etc. 
Wind-shakes especially affect the shearing strength. Also, if 
the required transverse strength is obtained by using high nar- 
row stringers, the area of pressure between the stringers and the 



§ 156. TRESTLES, 181 

cap niaj become so small as to induce crushing across the grain. 
This is a very common defect in trestle design. As already in- 
dicated in § 138, the span should vary roughly with the average 
hei^'ht of the trestle, the longer spans being employed when the 
trestle bents are very high, although it is usual to employ the 
same span throughout any one trestle. 

To illustrate, if we select a span of l-i feet, the load on one 
cap will be 74,200 lbs. If the stringers and cap are made of 
long-leaf yellow pine, which require the closely determined value 
of 1180 lbs. per square inch to produce a crushing amounting 
to ^fo of the height on timber with 12^ moisture, we may use 
200 lbs. per square inch as a safe pressure even for green tim- 
ber; this will require 371 square inches of surface. If the cap 
is 12^' wide, this will require a width of 31 inches, or say 2 
stringers under each rail, each 8 inches wide. For rectangular 
beams 

Moment = ^R'hh\ 

Using for 11' the safe value 1575 lbs. per square inch, we have 

142400 X 12 = i X 1575 X 32 X h\ 

from which h = 15^'. 9. If desired, the width may be increased 
to 9" and the depth correspondingly reduced, which will give 
similarly h = 14''. 8, or say 15". This show^s that two beams, 
9'' X 15'', under each rail will stand the transverse bending and 
have more than enough area for crushing. 
The shear per square inch will equal 

3 total shear 3 49600 ^^^., . ^ 

-. — = - zrz = 138 lbs. per so. inch, 

2 cross section 2 4 X 9 X lo ^ ^ 

which is a safe value, although it should preferably be less. 
Hence the above combination of dimensions will answer. 

The deflection should be computed to see if it exceeds the 



182 RAILROAD CONSTRUCTION, % 157. 

somewhat arbitrary standard of gi^ of the span. The deflection 
for tiniforni loading is 

A = 



Z^WE 



in which I = length in inches ; 

IF = total load, assumed as uniform ; 

E = modulus of elasticity, given as 2,070,000 lbs. 

per sq. in. for long-leaf pine, 12^ dry, and assumed to be 
1,200,000 for green timber. Then 

_ 5 X 72800 X 168 - 

~ 32 X 36 X 15" X 1200000 ~ 

^^X168''=0".84, 

so that the calculated deflection is well within the limit. Of 
course the loading is not strictly uniform, but even with a lib- 
eral allowance the deflection is still safe. 

For the heaviest practice (40000 lbs. per axle) these stringer 
dimensions must be correspondingly increased. 

157. Design of posts. Four posts are generally used for 
single-track work. The inner posts are usually braced by the 
cross-braces, so that their columnar strength is largely increased ; 
but as they are apt to get more than their share of work, the ad- 
vantage is compensated and they should be treated as unsupported 
columns for the total distance between cap and sill in simple 
bents, or for the height of stories in multiple-story construction. 
The caps and sills are assumed to have a width of 12''. It 
facilitates the application of bracing to have the columns of the 
same width and vary the other dimension as required. 

Unfortunately the experimental work of the U. S. Govern- 
ment on timber testing has not yet progressed far enough to 
establish unquestionably a general relation between the strength 
of lono- columns and the crushing: streno^th of short blocks. The 



§ 157. TRESTLES. 183 

following formula has been suggested, but it cannot be consid- 
ered as established : 

f = -F X »^r. ■ ^ ^ — . — -.^ in which 
J ^ '^ 700 + 15c + c'' 

f = allowable working stress per sq. in. for long columns; 
ji^== " '^ " " " " " short blocks; 

I 

I = length of column in inches ; 
d = least cross-sectional dimension in inches. 

Enough work has been done to give great reliability to the two 
following formulae for white pine and yellow pine, quoted from 
Johnson's '' Materials of Construction," p. 684 : 

1 fiy 

Working load per sq. in. =^ = 1000 — ^[jj ^ long-leaf pine; 

«< " '' " " =p = 600 — ^(y-j , white pine; 

in which I = length of column in inches, and 

h = least cross-sectional dimension in inches. 

The frequent practice is to use 12'' X 12'' posts for all tres- 
tles. If we substitute in the above formula ^ = 20' = 240" and 
h =. 12", we have p = 1000 - i(\V-)' = ^^0 lbs. 

900 X 1-11: = 129600 lbs., the loorking load for each post. 
This is more than the total load on one trestle bent and il- 
lustrates the usual great waste of timber. Making the post 
8" X 12" and calculating similarly, we have p = T75, and 
the working load per column is 775 X 06 = 74400 l])s. As 
considerable must be allowed for " weathering," which destroys 
the strength of the outer layers of the wood, and also for the 
dynamic effect of the live load, 8" X 12" may not be too great, 



184 BAILROAD CONSTRUCTION, § 158. 

but it is certainly a safe dimension. 12'' X 6" would possibly 
prove amply safe in practice. One method of allowing for 
weathering is to disregard the outer half-inch on all sides of the 
post, i.e., to calculate the strength of a post one inch smaller in 
each dimension than the post actually employed. On this basis 
an 8'' X 12'' X 20' post, computed as a 7" X 11' post, would 
have a safe columnar strength of 706 lbs. per square inch. With 
an area of 77 square inches, this gives a working load of 54362 
lbs. for each post^ or 217148 lbs. for the four posts. Consider- 
ing that 74200 lbs. is the maximum load on one cap (14 feet 
span), the great excess of strength is apparent. 

158. Design of caps and sills. The stresses in caps and sills 
are very indefinite, except as to crushing across the grain. As 
the stringers are placed almost directly over the inner posts, and 
as the sills are supported just under the posts, the transverse 
stresses are almost insignificant. In the above case four posts 
have an area of 4 X 12" X 8" = 384 sq. in. The total load, 
74200 lbs., will then give a pressure of 193 pounds per square 
inch, which is within the allowable limit. This one feature 
might require the use of 8" X 12" posts rather than 6" X 12" 
posts, for the smaller posts, although probably strong enough as 
posts, would produce an objectionably high pressure. 

159. Bracing. Although some idea of the stresses in the 
bracing could be found from certain assumptions as to wind- 
pressure, etc. , yet it would probably not be found wise to de- 
crease, for the sake of economy, the dimensions which practice 
has shown to be sufiicient for the work. The economy that 
would be possible would be too insignificant to justify any risk. 
Therefore the usual dimensions, given in §§ 139 and 140, should 
be employed. 



CHAPTEE Y. 



TUNNELS. 



SURVEYING. 

160. Surface surveys. As tunnels are always dug from each 
end and frequently from one or more intermediate shafts, it is 
necessary that an accurate surface survey should be made 
between the two ends. As the natural surface in a locality 
where a tunnel is necessary is almost invariably very steep and 
rough, it requires the employment of unusually refined methods 
of work to avoid inaccuracies. It is usual to run a line on the 
surface that will be at every point vertically over the center line 
of the tunnel. Tunnels are generally made straight unless 
curves are absolutely necessary, as curves add greatly to the 
cost. Fig. 85 represents roughly a longitudinal section of the 




^ ^^-IQOOD' H* 7000 '>t--""CUWJ \ TyJO- : ^000^ j -5000— ^ 

Fig. 85 —Sketch of Section of the Hoosac Tunnel. 

Hoosac Tunnel. Permanent stations were located at ^i, B^ C^ 
/>, E^ and F^ and stone houses were built at A^ B, C^ and 7>. 
These were located with ordinary field transits at first, and then 
all the points were placed as nearly as possible in one vertical 
plane by repeated trials and minute corrections, using a verv 
large specially constructed transit. The stations 7> and F weva 
necessary because E and A were invisible from (7 niid /?. 

185 



186 RAILROAD CONSTRUCTION. § 160. 

The alignment at A and ^ having been determined with great 
accuracy, the true ahgnment was easily carried into the tunnel. 
The relative elevations of A and E were determined with 
great accuracy. Steep slopes render necessary many settings 
of the level per unit of horizontal distance and require that the 
work be unusually accurate to obtain even fair accuracy per 
unit of distance. The levels are usually re-run many times 
until the probable error is a very small quantity. 

The exact horizontal distance between the two ends of the 
tunnel must also be known, especially if the tunnel is on a 
grade. The usual steep slopes and rough topography likewise 
render accurate horizontal measurements very difficult. Fre- 
quently when the slope is steep the measurement is best 
obtained by measuring along the slope and allowing for grade. 
This may be very accurately done by employing two tripods 
(level or transit tripods serve the purpose very well), setting 
them up slightly less than one tape-length apart and measuring 
iDetween horizontal needles set in wooden blocks inserted in the 
top of each tripod. The elevation of each needle is also 
observed. The true horizontal distance between two successive 
positions of the needles then equals the square root of the 
difference of the squares of the inclined distance and the differ- 
ence of elevation. Such measurements will probably be more 
accurate than those made by attempting to hold the tape 
horizontal and plumbing down with plumb-bobs, because (1) 
it is practically difficult to hold both ends of the tape truly 
horizontal; (2) on steep slopes it is impossible to hold the down- 
Mil end of a 100-foot tape (or even a 25-foot length) on a level 
with the other end, and the great increase in the number of 
applications of the unit of measurement very greatly increases 
the probable error of the whole measurement ; (3) the vibrations 
of a plumb-bob introduce a large probability of error in trans- 
ferring the measurement from the elevated end of the tape to 
the ground, and the increased number of such applications of 
the unit of measurement still further increases the probable 
error. 



^ 161. TUNNELS. 187 



c 



161. Surveying down a shaft. If a shaft is sunk, as at S^ 
Fig. 85, and it is desired to dig out the tunnel in both directions 
from the foot of the shaft so as to meet tlie headings from the 
outside, it is necessary to know, when at the bottom of the 
shaft, the elevation, alignment, and horizontal distance from 
each end of the tunnel. 

The elevation is generally carried down a shaft by means of 
a steel tape. This method involves the least number of ai)pli- 
cations of the unit of measurement and greatly increases the 
accuracy of the final result. 

The horizontal distance from each end may be easily trans- 
ferred down the shaft by means of a plumb-bob, using sonVe of 
the precautions described in the next paragraph. 1 

To transfer the alignment from the surface to the bottom of 
a shaft requires the highest skill because the shaft is always 
small, and to produce a line perhaps several thousand feet long 
in a direction given by two points 6 or 8 feet apart requires 
that the two points must be determined with extreme accuracy.. 
The eminently successful method adopted in the Hoosac Tunnel 
will be briefly described : Two beams were securely fastened 
across the top of the shaft (1030 feet deep), tlie beams being: 
plaged transversely to tlie direction of the tunnel and as far 
apart as possible and yet allow plumb-lines, hung from the 
intersection of each beam with the tunnel center line, to swing 
freely at the bottom of the shaft. These intersections of the 
beams with the center line were determined by averaging the 
results of a large number of careful observations for alio-nment. 
Two fine parallel wires, spaced about J^" apart, were then 
stretched between the beams so that the center line of the 
tunnel bisected at all points the space between the wires. 
Plumb-bobs, weighing 15 pounds, were suspended by fine wires 
beside each cross-beam, the wires passing between the two 
parallel alignment wures and bisecting the space. The plumb- 
bobs were allowed to swing in pails of water at the. bottom. 
Drafts of air up the shaft required the construction of boxes 
surrounding the wires. Even these precautions did not suffice 



188 RAILROAD CONSTRUCTION. § 162. 

to absolutely prevent vibration of the wire at the bottom 
through a very small arc. The mean point of these vibrations 
in each case was then located on a rigid cross-beam suitably 
placed at the bottom of the shaft and at about the level of the 
roof of the tunnel. Short plumb-lines were then suspended 
from these points whenever desired ; a transit was set (by trial) 
so that its line of collimation passed through both plumb lines 
and tlie line at the bottom could thus be prolonged. 

162. Underground surveys. Survey marks are frequently 
placed on the timbering, but they are apt to prove unreliable on 
account of the shifting of the timbering due to settlement of the 
surrounding material. They should never be placed at the bottom 
of the tunnel on account of the danger of being disturbed or 
covered up. Frequently holes are drilled in the roof and filled 
with wooden plugs in which a hook is screwed exactly on line. 
Although this is probably the safest method, even these plugs are 
not always undisturbed, as the material, unless very hard, will 
often settle slightly as the excavation proceeds. When a tunnel 
is perfectly straight and not too long, alignment-points may be 
given as frequently as desired from permanent stations located 
outside the tunnel where they are not liable to disturbance. 
This has been accomplished by running the alignment through 

the upper part of the cross-section, at 

one side of the center, where it is out of 

the way of the piles of masonry material, 

debris, etc., which are so apt to choke 

up the lower part of the cross-section. 

The position of this line relative to the 

cross-section being fixed, the alignment 

of any required point of the cross-section 

is readily found by means of a light frame 

or template with a fixed target located 

where this line would intersect the frame 

FjG. 86. when properly placed. A level -bubble 

on tlie frame will assist in setting the frame in its proper position. 

In all tunnel surveying the cross-wires must be illuminated 




§ 163. TUNNELS. 189 

by a lantern, and the object sighted at must also be illuminated. 
A powerful dark-lantern with the opening covered with (jroxind 
glass has been found useful. This may be used to illuminate a 
plumb-bob string or a very fine rod, or to place behind a brass 
plate having a narrow slit in it, the axis of the slit and plate 
being coincident with the plumb-bob string by which it is hung. 

On account of the interference to the surveying caused by 
the work of construction and also by the smoke and dust in the 
air resulting from the blasting, it is generally necessary to make 
the surveys at times when construction is tenq^orarily sus- 
pended. 

163. Accuracy of tunnel surveying. Apart from the very 
natural desire to do surveying which shall check well, there is 
an important financial side to accurate tunnel surveying. If 
the survev lines do not meet as desired when the headino^s come 
together, it may be found necessary, if the error is of appreciable 
size, to introduce a slight curve, perhaps even a reversed curve, 
into the alignment, and it is even conceivable that the tunnel 
section would need to be enlarged somewhat to allow for these 
curves. The cost of these changes and the perpetual annoyance 
due to an enforced and undesirable alteration of the original 
design will justify a considerable increase in the expenses of the 
survey. Considering that the cost of surveys is usually but a 
small fraction of the total cost of the work, an increase of 10 or 
even 20^ in the cost of the surveys will mean an insignificant 
addition to the total cost and frequently, if not generally, it will 
result in a saving of many times the increased cost. The 
accuracy actually attained in two noted American tunnels is 
given as follows : The Musconetcong tunnel is about 5000 feet 
long, bored through a mountain -100 feet high. The error of 
alignment at the meeting of the headings was O'.Oi, error of 
levels O'.Olo, error of distance 0'.52. The Hoosac tunnel is 
over 25,000 feet long. The heading from the east end met the 
heading from the central shaft at a point 11271 feet from the 
east end and 15G3 feet from the shaft. The error in align- 
ment was y\ of an inch, that of levels " a few hundredths," 



190 RAILROAD CONSTRUCTION. % 164. 

error of distance ' ' trifling. ' ' Tlie alignment, corrected at the 
shaft, was carried on through and met the heading from the west 
end at a point 10138 feet from the west end and 2056 feet from 
the shaft. Here the error of alignment was -^^" and that of 
levels 0.134 ft. 



DESIGN. 

164. Cross-sections, l^early all tunnels have cross-sections 
peculiar to themselves — all varying at least in the details. The 
general form of a great many tunnels is that of a rectangle sur- 
mounted by a semi-circle or semi-ellipse. In very soft material 
an inverted arch is necessary along the bottom. In such cases 
the sides will generally be arched instead of vertical. The sides 
are frequently battered. With very long tunnels, several forms 
of cross-section will often be used in the same tunnel, owing to 
differences in the material encountered. In solid rock, which 
w^ill not disintegrate upon exposure, no lining is required, and 
the cross-section will be the irregular section left by the blasting, 
the only requirement being that no rock shall be left within the 
required cross-sectional figure. Farther on, in the same tunnel, 
when passing through some very soft treacherous material, it 
may be necessary to put in a full arch lining — top, sides, and bot- 
tom — which will be nearly circular in cross-section. For an 
illustration of this see Figs. 87 and 88. 

The width of tunnels varies as greatly as the designs. Single- 
track tunnels generally have a width of 15 to 16 feet. Occa- 
sionally they have been built 14 feet wide, and even less, and 
also up to 18 feet, especially when on curves. 24 to 26 feet is 
the most common width for double track. Many double- track 
tunnels are only 22 feet wide, and some are 28 feet wide. The 
heights are generally 19 feet for single track and 20 to 22 feet 
for double track. The variations from these figures are con- 
siderable. The lower limits depend on the cross-section of the 
rolling stock, with an indefinite allowance for clearance and ven- 
tilation. Cross-sections which coincide too closely with what is 



164. 



TUNNELS. 



191 




Fig. 87.— Housac Tunnel. Section through Solid Rock. 




Fig. 88.— Hoosac Tunnel Section through Soft Ground, 



192 



RAILROAD CONSTRUCTION. 



165. 



absolutely required for clearance are objectionable, because any 
slight settlement of the lining which would otherwise be harm- 
less would then become troublesome and even dangerous. Figs. 
87, 88, and 89 ^ show some typical cross-sections. 




Fig. 89. — St. Cloud Tunnel. 

165. Grade. A grade of at least 0.2^ is needed for drainage. 
If the tunnel is at the summit of two grades, the tunnel grade 
should be practically level, with an allowance for drainage, the 
actual summit being perhaps in the center so as to drain both 
ways. When the tunnel forms part of a long ascending grade, 
it is advisable to reduce the grade through the tunnel unless the 
tunnel is very short. The additional atmospheric resistance and 
the decreased adhesion of the driver wheels on the damp rails in 
a tunnel will cause an engine to work very hard and still more 
rapidly vitiate the atmosphere until the accumulation of poison- 
ous gases becomes a source of actual danger to the engineer and 
fireman of the locomotive and of extreme discomfort to the 
passengers. If the nominal ruling grade of the road were 
maintained through a tunnel, the maximum resistance would be 



* Drinker's "Tunneling. 



PLATE 11. 




TuN^'El.-TIMBERI^G— English System (6*). 




TUNNEL-TIMBERTXG— ENGTJSn SYSTEM (6). 

{To face 'page 192.) 



PLATE III. 



■///■//■/r.'Mi'i)j'''',''i:/M'/'ir'//.'ii//.i//,'///f,/j'/'//'-^^i ', /, • .w ;^y:'y/<>-y//fr/M/',r,"'^////'y, ///i-'^y^v/fif^y^ 




TUJS2HiiL-TIMJ3EKlIS'G— ENGLISH SYSTEM {c). 




TuNNEL-TTMBERiKG— English System {d), 
{To face page 192.) 



§ 167. TUNNELS. 193 

found in the tunnel. This would probably cause trains to stall 
there, which would be objectionable and perhaps dangerous. 

166. Lining. It is a characteristic of many kinds of rock 
and of all earthy material that, although they may be self-sus- 
taining when first exposed to the atmosphere, they rapidly dis- 
integrate and require that the top and perhaps the sides and 
even the bottom shall be lined to prevent caving in. In this 
country, when timber is cheap, it is occasionally framed as an 
arch and used as the permanent lining, but masonry is always 
to be preferred.- Frequently the cross-section is made extra 
large so that a masonry lining may subsequently be placed inside 
the wooden lining and thus postpone a large expense until the 
road is better able to pay for the work. In very soft unstable 
material, like quicksand, an arch of cut stone voussoirs may be 
necessary to withstand the pressure. A good quality of brick is 
occasionally used for lining, as they are easily handled and make 
good masonry if the pressure is not excessive. Only the best 
of cement mortar should be used, economy in this feature being 
the worst of folly. Of course the excavation must include the 
outside line of the lining. Any excavation which is made out- 
side of this line (by the fall of earth or loose rock or by exces- 
sive blasting) must be refilled with stone well packed in. Occa- 
sionally it is necessary to fill these spaces with concrete. Of 
course it is not necessary that the lining be uniform throughout 
the tunnel. 

167. Shafts. Shafts are variously made with square, rectan- 
gular, elliptical, and circular cross-sections. The rectangular 
cross-section, with the longer axis parallel with the tunnel, is 
most usually employed. Generally the shaft is directly over the 
center of the tunnel, but that always implies a complicated con- 
nection between the linings of the tunnel and shaft, provided 
such linings are necessary. It is easier to sink a shaft near to 
one side of the tunnel and make an opening through the nearly 
vertical side of the tunnel. Such a method was employed in the 
Church Hill Tunnel, illustrated in Fig. 90.'' Fig. 91 f shows 

* Drinker's "Tunneling." 

f Rziha, " Lebrbuch der Gesammteu Tunnelbaukunsi." 



194 



RAILROAD CONSTRUCTION, 



167. 



a cross-section for a large main sliaft. Many shafts have been 
built with the idea of being left open permanently for ventila- 
tion and have therefore been elaborately Uned with masonry. 




Fig. 90.— Connection with Shaft, Church Hill Tunnel. 




Fig. 91. — Cross-section, Large Main Shaft. 

The general consensus of opinion now appears to be that shafts 
are worse than useless for ventilation ; that the quick passage of 
a train through the tunnel is the most effective ventilator ; and 
that shafts only tend to produce cross-currents and are ineffective 
to clear the air. In consequence, many of these elaborately 
lined shafts have been permanently closed, and the more recent 



PLATE IV 







TuNNEL-TiMBEKiNG— French System (a). 




TuNN^:L-TIMREHI^•G-FKE^XII System (&). 
(To face page 194.) 



PLATE y. 




Tunnel timbeking— Belgian Sy&tem (a). 




Mi 



Tunnel-timbeking— Belglvn System (6). 
{To face 'page 194.) 



§ 169. TUNNELS. 195 

practice is to close up a shaft as soon as the tunnel is completed. 
Shafts always form drainage-wells for the material they pass 
through, and sometimes to such an extent that it is a serious 
matter to dispose of the water that collects at the bottom, 
requiring the construction of large and expensive drains. 

168. Drains. A tunnel will almost invariably strike veins of 
water which will promptly begin to drain into the tunnel and 
not only cause considerable trouble and expense during construc- 
tion, but necessitate the provision of permanent drains for its 
perpetual disposal. These drains nmst frequently be so large as 
to appreciably increase the required cross- section of the tunnel. 
Generally a small open gutter on each side will suffice for this 
purpose, but in double-track tunnels a large covered drain is 
often built between the tracks. It is sometimes necessary to 
thoroughly grout the outside of the lining so that water will not 
force its way through the masonry and perhaps injure it, but 
may freely drain down the sides and pass through openings in 
the side walls near their base into the gutters. 

CONSTRUCTION. 

169. Headings. The methods of all tunnel excavation de- 
pend on the general principle that all earthy material, except 
the softest of liquid mud and quicksand, will be self-sustaining 
over a greater or less area and for a greater or less time after 
excavation is made, and the work consists in excavating some 
material and immediately propping up the exposed surface by 
timbering and poling-boards. The excavation of the cross-sec- 
tion begins with cutting out a ''heading," which is a small 
horizontal drift whose breast is constantly kept 15 feet or 
more in advance of the full cross-sectional excavation. In solid 
self-sustaining rock, which will not decompose upon exposure 
to air, it becomes simply a matter of excavating the rock with 
the least possible expenditure of time and energy. In soft 
ground the heading must be heavily timbered, and as the heading 
is gradually enlarged the timbering must be gradually extended 



196 



RAILROAD CONSTRUCTION, 



170. 



and perhaps replaced, according to some regular system, so that 
when the full cross-section has been excavated it is supported 
by such timbering as is intended for it. The heading is some- 
times made on the center line near the top ; with other plans, 
on the center line near the bottom ; and 
sometimes two simultaneous headings are run 
in the two low^er corners. Headings near the 
bottom serve the purpose of draining the 
material above it and facilitating the excava- 
tion. The simplest case of heading timber- 
ing is that shown in Fig. 92, in which cross- 
timbers are placed at intervals just under the 
roof, set in notches cut in the side walls and 
supporting poling-boards which sustain what- 
ever pressure may come on them. Cross-timbers near the bottom 
support a flooring on which vehicles for transporting material 
may be run and under which the drainage may freely escape. 
As the necessity for timbering becomes greater, side timbers and 
even bottom timbers must be added, these timbers supporting 
poHng-boards, and even the breast of the heading must be pro- 
tected by boards suitably braced, as shown in Fig. 93. The 




Fig. 93. 




Fk;} 93 — TiMBEEixa FOK Tunnel Heading. 

supporting timbers are framed into collars in such a manner that 
added pressure only increases their rigidity. 

170. Enlargement. Enlargement is accomplished by remov- 
ing the poling-boards, one at a time, excavating a greater or less 



PLATE VI. 



-■' '7^^ ;~^^ ■'////'"/ j'^^>-4' •: " 



















•^^ -.^./ / 



Tunnel timeeiung— German System (a). 



''':■'::: -^:';:>,'r.,Xir\ 



•^v/'/'^ii', 










■^- 



>ci 



v^ 






^^^;^^: ., ::-^^:^-.;^J^.i•'^- 



;'^^"■ 



''^-'^^^Ifv^ :^?^^ 



TuNNEL-TiMBET? TNG— German Sys'iem (6). 
{To face page 196.) 



PLATE VII. 




pr-T^^i^ 






-^/^" 














Tunnel-timbering — German System [c). 



JmiW^r^''^^ 




•^' 






Tunnel-timbering— German System {d) 
{To face page 1% ) 



§171. 



TUNNELS. 



197 



amount of material, and immediately supporting the exposed 
material with poling-boards suitably braced. (See Figs. 93 and 
94.) This work being systematically done, space is therel)y 




Fig. 94. 



obtained in which the framing for the full cross- section may be 
gradually introduced. The framing is constructed with a cross- 
section so large that the masonry lining may be constructed 
within it. 

171. Distinctive features of various methods of construction. 
There are six general systems, known as the English, German, 
Belgian, French, Austrian, and American. They are so named 
from the origin of the methods, although their use is not confined 
to the countries named. Fig. 95 shows by numbers (1 to 5) 
the order of the excavation within the cross-sections. The Eriir- 
lish, Austrian, and American systems are alike in excavating the 
entire cross-section before beginning the construction of the 
masonry lining. The German method leaves a solid core (5) 
until practically the whole of the lining is complete. This has 
the disadvantage of extremely cramped quarters for work, poor 
ventilation, etc. The Belgian and French methods agree in 
excavating the upper part of the section, building the arch at 
once, and supporting it temporarily until the side walls are 
built. The Belgian method then takes out the core (3), removes 
very short sections of the sides (4), immediately underpinning 
the arch with short sections of the side walls and thus gradually 
constructing the whole side wall. The French method digs out 
the sides (3), supporting the arch temporarily with timbers and 



198 



RAILROAD CONSTRUCTION. 



171. 



tlien replacing the timbers with masonry ; the core (4) is taken 
out last. The French method has the same disadvantage as the 
German — working in a cramped space. The Belgian and French 
systems have the disadvantage that the arch, supported tempo- 
rarily on timber, is very apt to be strained and cracked by the 
slight settlement that so frequently occurs in soft material. The 
English, Austrian, and American methods differ mainly in the 



f' 


^1 
1 

1 

1 
1 

1 
1 

-+- 

1 
1 
1 

-+- 

1 

1 


1 


fx 


^ 


4 


3 


-4-- 


4 


5 


1 


5 





ENGLISH 



AUSTRIAN 



AMERICAN 






german belgian french 

Fig. 95. — Order of Working by the Various Systems. 

design of the timbering. The English support the roof by lines 
of very heavy longitudinal timbers which are supported at com- 
paratively wide intervals by a heavy framework occupying the 
whole cross-section. The Austrian system uses such frequent 
cross-frames of timber- work that poling-boards will suffice to 
support the material between the frames. The American sys- 
tem agrees with the Austrian in using frequent cross-frames 
supporting poling-boards, but differs from it in that the '' cross- 
frames " consist simply of arches of 3 to 15 wooden voussoirs, 
the voussoirs being blocks of 12" X 12'' timber about 2 to 8 feet 
long and cut with joints normal to the arch. These, arches are 
put together on a centering which is removed as soon as the arch 



PLATE VIII. 







TUNNEL-TIMBEIIING — AUSTRIAN SYSTEM (a). 




TUNNEL-TlMliEKlNG — AUSTRIAN JSYSTEM (6). 




TuNNET,-TiMBERiNG— Austrian System (rt). 
{To face page 198.) 



PLATE IX. 



^ 






MM^ 



^'^ 



.0 






\\' 






TuNNEL-TiMiifiiiiNa— Austrian System [d). 













-^-<v.''J^^ 



Tunnel-timbering — Austrian System {e). 
(To face page 198.) 



PLATE X. 







Tunnel-timbering— AusTiaAN System (/). 



,^ ^^ V V ^^"^ --^"^ ^'^ vi.-^^:::^^ . 




Tunnel-timbering — Austrian System {g). 
(To face page 198 ) 



g 173. TUNNELS. 199 

is keyed up and thus immediately opens up the full cross-section, 
so that the center core (4) may be immediately dug out and the 
masonry constructed in a large open space. The American sys- 
tem has been used successfully in very soft ground, but its ad- 
vantages are greater in loose rock, when it is much cheaper than 
the other methods which employ more timber. Fig. 90 illus- 
trates the use of the American system. The iigure shows the 
wooden arch in place. The masonry arch may be placed when 
convenient, since it is possible to lay the track and commence 
traffic as soon as the wooden arch is in place. Plates II to XIV 
illustrate the methods of excavating and timbering by these 
various systems. 

172. Ventilation during construction. Tunnels of any great 
length must be artificially ventilated during construction. If 
the excavated material is rock so that blasting is necessary, the 
need for ventilation becomes still more imperative. The inven- 
tion of compressed-air drills simultaneously solved two difficul- 
ties. It introduced a motive power which is unobjectionable in 
its application (as gas would be), and it also furnished at the same 
time a supply of just what is needed — pure air. If no blasting 
is done (and sometimes even when there is blasting), air must be 
supplied by direct pumping. The cooling effect of the sudden 
expansion of compressed air only reduces the otherwise objection- 
ably high temperature sometimes found in tunnels. Since pure 
air is being continually pumped in, the foul air is thereby forced 
out. 

173. Excavation for the portals. Under normal conditions 
there is always a greater or less amount of open cut preceding 
and following a tunnel. Since all tunnel methods depend (to 
some slight degree at least) on the capacity of the exposed ma- 
terial to act as an arch, there is implied a considerable thickness 
of material above the tunnel. This thickness is reduced to 
nearly zero over the tunnel portals and therefore requires special 
treatment, particularly when the material is very soft. Fig. 90 * 

* Kziba, " Lehrbucli der Qersammten Tunnelbaukunst." 



200 



BAILROAD CONSTRUCTION, 



174. 



illustrates one method of breaking into the ground at a portal. 
The loose stones are piled on the framing to give stability to the 
framing by their weight and also to retain the earth on the 
slope above. Another method is to sink a temporary shaft to 
the tunnel near the portal ; immediately enlarge to the full size 
and build the masonry lining ; then work back to the portal. 




Fig. 96.— Timbering for Tunnel Portal. 

This method is more costly, but is preferable in very treacherous 
ground, it being less liable to cause landslides of the surface 
material. 

174. Tunnels vs. open cuts. In cases in which an open cut 
rather than a tunnel is a possibility the ultimate consideration 
is generally that of first cost combined with other financial con- 
siderations and annual maintenance charges directly or indirectly 
connected with it. Even when an open cut may be constructed at 
the same cost as a tunnel (or perhaps a little cheaper) the tunnel 
may be preferable under the following conditions : 

1 . When the soil indicates that the open cut would be liable 
to landslides. 

2. When the open cut would be subject to excessive snow- 
drifts or avalanches. 

3. When land is especially costly or it is desired to run under 
existing costly or valuable buildings or monuments. When run- 
ning through cities, tunnels are sometimes constructed as open 
cuts and then arched over. 



PLATE XL 





/yYZ. 






PERMANENT TIMBERING OF HEADING. \ \ 

\ \ 



/ 



j- _ ] \ 

', t r 

PncENixviLLE Tunnel. P. S. V. R.R. 
{To face page 200.) 



PLATE XIL 





Phcenixville Tunnel, P. S. V. R.R. 
{To face page %<dO.) 



PLATE XIII. 




Phcenixville Tunnel. P. S. V. R.R. 



{To face page 2^0.) 



PLx\TE XIV. 




Elevatio:^ of Purtai., 




Longitudinal Section of Portal. 
PnoENixviLLE Tunnel. P. S. V. R.R. 
iTo face 'page 2^<^.) 



§175. 



TUN2iELS. 



201 



These cases apply to tunnels vs. open cuts when the ahgn- 
ment is fixed by other considerations than the mere topography. 
The broader question of excavating tunnels to avoid excessive 
grades or to save distance or curvature, and siniihir problems, 
are hardly susceptible of general analysis except as questions of 
railway economics and must be treated individually. 

175. Cost of tunneling. Tlie cost of any construction wdiicli 
involves such uncertainties as tunneling is very varinble. It de- 
pends on the material encountered, the amount and kind of tim- 
bering required, on the size of the cross- section, on the price of 
labor, and especially on the reconstruction that may be necessary 
on account of mishaps. 

Headings generally cost $4 to $5 per cubic yard for excava- 
tion, while the remainder of the cross-section in the same tunnel 
may cost about half as much. The average cost of a large number 
of tunnels in this country may be seen from the following table : * 



Material. 


Cost per cubic yard. 


Cost per 
lineal foot. 


Excavation. 


Masonry. 




Single. 


Double. 


Single. 


Double. 


Single. 


Double. 


Hard rock 

Loose rock. . . . 
Soft ground. . . 


$5.89 
3.12 
3.62 


$5.45 
3.48 
4.64 


$12.00 

9.07 

15.00 


$ 8.25 
10.41 
10.50 


% 69.76 

80.61 

135.31 


$142.82 
119.26 
174.42 



A considerable variation from these figures may be found in 
individual cases, due sometimes to unusual skill (or the lack of 
it) in prosecuting the work, but the figures will generally be 
sufficiently accurate for preliminary estimates or for the compari- 
son of two proposed routes. 

* Figures derived from Drinker's "Tunneling." 



CHAPTEK YI. 

CULVERTS AND MINOR BRIDGES. 

176. Definition and object. Although a variable percentage 
of the rain falling on any section of country soaks into the 
ground and does not immediately reappear, yet a very large 
percentage flows over the surface, always seeking and following 
the lowest channels. The roadbed of a railroad is constantly 
intersecting these channels, which frequently are normally drv. 
In order to prevent injury to railroad embankments by the im- 
pounding of such rainfall, it is necessary to construct waterways 
through the embankment through which such rainflow may 
freely pass. Such waterways, called culverts, are also appli- 
cable for the bridging of very small although perennial streams, 
and therefore in this work the term culvert will be applied to 
all water-channels passing through a railroad embankment which 
are not of sufficient magnitude to require a special structural 
design, such as is necessary for a large masonry arch or a truss 
bridge. 

177. Elements of the design. A well-designed culvert must 
afford such free passage to the water that it will not ' ' back up ' ' 
over the adjoining land nor cause any injury to the embankment 
or culvert. The ability of the culvert to discharge freely all the 
water that comes to it evidently depends chiefly on the area of 
the waterway, but also on the form, length, slope, and materials 
of construction of the culvert and the nature of the approach 
and outfall. When the embankment is very low and the amount 
of water to be discharged very great, it sometimes becomes 
necessary to allow the water to discharge " under a head," i.e., 

202 



§ 178. CULVERTS AND MINOR BRIDGES. 203 

with the surface of the water above the top of the culvert. 
Safety then requires a much stronger construction than would 
otherwise be necessary to avoid injury to the culvert or embank- 
ment by wasliing. The necessity for such construction should 
be avoided if possible. 

AREA OF THE WATERWAY. 

178. Elements involved. The determination of the required 
area of the waterway involves such a nmltiplicity of indeter- 
minate elements that any close determination of its value from 
purely theoretical considerations is a practical impossibility. 
The principal elements involved are: 

a. Rainfall. The real test of the culvert is its capacity to 
discharge without injury the flow resulting from the extraordi- 
nary rainfalls and "cloud bursts" that may occur once in many 
years. Therefore, while a knowledge of the average annual 
rainfall is of very little value, a record of the maximum rainfall 
during heavy storms for a long term of years may give a relative 
idea of the maximum demand on the culvert. 

b. Area of watershed. This signifies the total area of country 
draining into the channel considered. When the drainage 
area is very small it is sometimes included within the area 
surveyed by the preliminary survey. When larger it is fre- 
quently possible to obtain its area from other maps with a per- 
centage of accuracy sufficient for the purpose. Sometimes a 
special survey for the purpose is considered justifiable. 

c. Character of soil and vegetation. This has a large in- 
fluence on the rapidity with which the rainflow from a given 
area will reach the culvert. If the soil is hard and impermeable 
and the vegetation scant, a heavy rain will run off suddenly, 
taxing the capacity of the culvert for a short time, while a 
spongy soil and dense vegetation will retard the flow, making it 
more nearly uniform and the maximum flow at any one time 
much less. 

d. Shape and slope of watershed. If the watershed is very 
long and narrow (other things being equal), the water from the 



204 RAILROAD COJSSTRUCTION. % 179. 

remoter parts will require so mucli longer time to reach the 
<;ulvert that the flow will be comparatively uniform, especially 
when the sloj)e of the whole watershed is very low. When the 
slope of the remoter portions is quite steep it may result in the 
nearly simultaneous arrival of a storm-flow from all parts of the 
watershed, thus taxing the capacity of the culvert. 

e. Effect of design of culvert. The principles of hydraulics 
show that the slope of the culvert, its length, the form of the 
cross-section, the nature of the surface, and the form of the 
approach and discharge all have a considerable influence on the 
area of cross-section required to discharge a given volume of 
water in a given time, but unfortunately the combined 
hydraulic efi^ect of these various details is still a very uncertain 
quantity. 

179. Methods of computation of area. There are three pos- 
sible methods of computation. 

(a) Theoretical. As shown above it is a practical impossi- 
bilitv to estimate correctlv the combined eftect of the ^reat mul- 
tiplicity of elements which influence the final result. The nearest 
approach to it is to estimate by the use of empirical formulae 
the amount of water which will be presented at the upper end 
of the culvert in a given time and then to compute, from the 
principles of hydraulics, the rate of flow through a culvert of 
given construction, but (as shown in § 178, e) such methods are 
still very unreliable, owing to lack of experimental knowledge. 
This method has apparently greater scientific accuracy than other 
methods, but a little study will show that the elements of un- 
certainty are as great and the final result no more reliable. The 
method is most reliable for streams of uniform flow, but it is 
under these conditions that method (c) is most useful. The 
theoretical method will not therefore be considered further. 

(b) Empirical. As illustrated in § 180, some formulae make 
the area of waterway a function of the drainage area, the for- 
mula being afl'ected by a coeflicient the value of which is esti- 
mated between limits according to the judgment. Assuming 
that the formulae are sound, their use only narrows the limits of 



§ 180. CULVERTS AND MINOR BRIDGES. 205 

error, the final determination depending on experience and judg- 
ment. 

(c) From observation. This method, considered by far the 
best for permanent work, consists in observing the high -water 
marks on contracted channel-openings which are on the same 
stream and as near as possible to the proposed culvert. If the 
country is new and there are no such openings, the wisest plan 
is to bridge the opening by a temporary structure in wood which 
has an ample waterway (see § 126, J, 4) and carefully observe 
all high- water marks on that opening during the 6 to 10 years 
which is ordinarily the minimum life of such a structure. As 
shown later, such observations may be utilized for a close com- 
putation of the required waterway. Method (b) may be utilized 
for an approximate calculation for the required area for the tem- 
porary structure, using a value which is intentionally excessive, 
so that a permanent structure of sufficient capacity may subse- 
quently be constructed within the temporary structure. 

180. Empirical formulae. Two of the best known empirical 
formulae for area of the waterway are the following : 

(a) Myer's formula: 

Area of waterway in square feet = C X V^drainage area in acres, 
where 6^ is a coefficient varying from 1 for flat country to 4 for 
mountainous country and rocky ground. As an illustration, if 
the drainage area is 100 acres, the waterway area should be from 
10 to 40 square feet, according to the value of the coefficient 
chosen. It should be noted that this formula does not reo^ard 
the great variations in rainfall in various parts of the world nor 
the design of the culvert, and also that the final result depends 
largely on the choice of the coefficient. 

(b) Talbot's formula: 

Area of waterway in square feet = C X V(drainage area in acres)'. 
" For steep and rocky ground O varies from | to 1. For rolling 
agricultural country subject to floods at times of melting snow, 
and with the length of the valley three or four times its width, 
is about J; and if the stream is longer in proportion to the area, 
decrease 0. In districts not affected by accumulated snow, and 



206 RAILROAD CONSTRUCTION, § 181. 

where the length of the valley is several times the width, 1 or i 
or even less, may be used. C should be increased for steep side 
slopes, especially if the upper part of- the valley has a much 
greater fall than the channel at the culvert. " ^ As an illustration, 
if the drainage area is 100 acres the area of waterway should be 
Cx 31.6. The area should then vary from 5 to 31 square 
feet, according to the character of the country. Like the 
previous estimate, the result depends on the choice of a coef- 
ficient and disregards local variations in rainfall, except as they 
may be arbitrarily allowed for in choosing the coefficient. 

181. Value of empirical formulse. The fact that these for- 
mulae, as well as many others of similar nature that have been 
suggested, depend so largely upon the choice of the coefficient 
shows that they are valuable " more as a guide to the judgment 
than as a working rule," as Prof. Talbot exphcitly declares in 
commenting on his own formula. In short, they are chiefly valu- 
able in indicating a probable maximum and minimum between 
which the true result probably lies. 

182. Results based on Observation. As already indicated in 
§ 179, observation of the stream in question gives the most 
reliable results. If the country is new and no records of the 
flow of the stream during heavy storms has been taken, even 
the life of a temporary wooden structure may not be lono- 
enough to include one of the unusually severe storms which 
must be allowed for, but there will usually be some high-water 
mark which will indicate how much opening will be required. 
The following quotation illustrates this: "A tidal estuary may 
generally be safely narrowed considerably from the extreme 
water lines if stone revetments are used to protect the 
bank from wash. Above the true estuary, where the stream 
cuts through the marsh, we generally find nearly vertical banks, 
and we are safe if the faces of abutments are placed even with 
the banks. In level sections of the country, where the current 
is sluggish, it is usually safe to encroach somewhat on the 

*Prof. A. N. Talbot, "Selected Papers of the Civil Engineers' Club of 
the Univ. of Illinois." 



§ 183. CULVERTS AND MINOR BRIDGES. 207 

general width of the stream, but in rapid streams among the 
hills the width that the stream has cut for itself through the 
soil should not be lessened, and in ravines carrying mountain 
torrents the openings must be left very much larger than the 
ordinary appearance of the banks of the stream would seem to 
make necessary." " 

As an illustration of an observation of a storm-flow throuo^h 
a temporary trestle, the following is quoted: ^'Having the 
flood height and velocity, it is an easy matter to determine the 
volume of water to be taken care of. I have one ten-bent pile 
trestle 135 feet long and 24 feet high over a spring branch that 
ordinarily runs about six cubic inches per second. Last sum- 
mer during one of our heavy rainstorms (four inches in less 
than three hours) I visited this place and found by float observa- 
tions the surface velocity at the highest stage to be 1.9 feet per 
second. I made a high- water mark, and after the flood- water 
receded found the width of stream to be 12 feet and an averao^e 
depth of 2} feet. This, with a surface velocity of 1.9 feet i)er 
second, would give approximately a discharge of 50 cubic feet, 
or 375 gallons, per second. Having this information it is easy 
to determine size of opening required." f 

183. Degree of accuracy required. The advantages result- 
ing from the use of standard designs for culverts (as well as 
other structures) have led to the adoption of a comparatively 
small number of designs. The practical use made of a compu- 
tation of required waterway area is to determine which one of 
several standard designs will most nearly fulflll the require- 
ments. For example, if a 24-incli iron pipe, having an area of 
3.14 square feet, is considered to be a little small, the next size 
(30-inch) would be adopted ; but a 30-inch pipe has an area of 
4.92 square feet, which is ^Q% larger. A siniihir result, except 
that the percentage of difference might not be quite so marked, 

* J. P. Snow, Boston & Maine Railway. From Report to Association of 
Railway Superintendents of Bridges and Buildings. 1897. 

f A. J. Kelley, Kansas City Belt Railway. From Report to Association 
of Railway Superintendents of Bridges and Buildings. 1807. 



S08 RAILROAD CONSTRUCTION, § 184. 

will be found by comparing the areas of consecutive standard 
designs for stone box culverts. 

The advisability of designing a culvert to withstand any 
storm-flow that may ever occur is considered doubtful. Several 
years ago a record-breaking storm in New England carried 
away a very large number of bridges, etc., hitherto supposed to 
be safe. It was not afterward considered that the design of 
those bridges was faulty, because the extra cost of constructing 
bridges capable of withstanding such a flood, added to interest 
for a long period of years, would be enormously greater than 
the cost of repairing the damages of such a storm once or twice 
in a century. Of course the element of danger has some 
weight, but not enough to justify a great additional expendi- 
ture, for common prudence would prompt unusual precautions 
during or immediately after such an extraordinary storm. 

PIPE CULVERTS. 

184. Advantages. Pipe culverts, made of cast iron or 
earthenware, are very durable, readily constructed, moderately 
cheap, will pass a larger volume of water in proportion to the 
area than many other designs on account of the smoothness of 
the surface, and (when using iron pipe) may be used very close 
to the track w^hen a low opening of large capacity is required. 
Another advantage lies in the ease with which they may be in- 
serted through a somewhat larger opening that has been tem- 
porarily lined with wood, without disturbing the roadbed or 
track. 

185. Construction. Permanency requires that the founda- 
tion shall be firm and secure against being washed out. To 
accomplish this, the soil of the trench should be hollowed out to 
fit the lower half of the pipe, making suitable recesses for the 
bells. In very soft treacherous soil a foundation -block of con- 
crete is sometimes placed under each joint, or even throughout 
the whole length. When pipes are laid through a sHghtly 
larger timber culvert great care should be taken that the pipes 
are properly supported, so that there will be no settling nor 



§ 186. CUL VERTS AND MINOR BRIDGES. 209 

development of unusual strains when tlie timber finally decays 
and gives way. To prevent the washing away of material 
around the pipe tlie ends should be protected by a bulkhead. 
This is best constructed of masonry (see Fig. 97), although wood 
is sometimes used for cheap and minor constructions. The joints 
should be calked, especially when the culvert is liable to run 
full or when the outflow is impeded and the culvert is liable to 
be partly or wholly tilled during freezing weather. The cost of 
a calking of clay or even hydraulic cement is insignificant com- 
l)ared with the value of the additional safety afforded. When 
the grade of the pipe is perfectly uniform, a very low rate of 
grade will suffice to drain a pipe culvert, but since some uneven- 
ness of grade is inevitable through uneven settlement or im- 
2)erfect construction, a grade of 1 in 20 should preferably be 
required, although much less is often used. The length of a 
pipe culvert is approximately determined as follows : 

Length = 25 {dejjth of embankment to top of pipe) + {width of roadbed)^ 

in which s is the slope ratio (horizontal to vertical) of the banks. 
In ])ractice an even number of lengths will be used which will 
most nearly agree with this formula. 

186. Iron-pipe culverts. Simple cast-iron pipes are used in 
sizes from 12'' to 48'' diameter. These are usually made in 
lengths of 12 feet w^th a few lengths of 6 feet, so that any 
required length may be more nearly obtained. The lightest 
pipes made are sufficiently strong for the purpose, and even those 
wdiicli would be rejected because of incapacity to withstand pres- 
sure may be utilized for this work. In Fig. 97 are shown the 
standard plans used on the C. C. C. & St. L. Ry., which may 
be considered as typical plans. 

Pipes formed of cast-iron segments liave been used up to 12 
feet diameter. The shell is then made comparatively thin, but 
is stiffened by ribs and flanges on the outside. The segments 
break joints and are bolted together through the flanges. The 
joints are made tight by the use of a tarred rope, together with 
neat cement. 



210 



RAILROAD CONSTRUCTION, 



§186. 




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k 








c 


3 






t"^^ 








4- 




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: 


^£ nvhIl J 


t 





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ss 


31 X 


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§187. 



CULVERTS AND MINOR BRIDGES. 



211 



187. Tile-pipe culverts. The pipes used for tliis purpose 
vary from 12" to 24" in diameter. When a larger capacity 
is required two or more pipes may be laid side by side, but in 
such a case another design might be preferable. It is frequently 
specified that ^ ' double-strength " or " extra-heavy ' ' pipe shall 
be used, evidently with the idea that the stresses on a culvert- 
pipe are greater than on a sewer-pipe. But it has been con- 
clusively demonstrated that, no matter how deep the embankment, 
the pressure cannot exceed a somewhat uncertain maximum, 
also that the greatest danger consists in placing the pipe so near 
the ties that shocks may be directly transferred to the pipe with- 
out the cusliioning effect of the earth and ballast. When the 
pipes are well bedded in clear earth and there is a sufficient 
depth of earth over them to avoid direct impact (at least three 
feet) the ordinary sewer-pipe will be sufficiently strong. 
^'Double-strength " pipe is frequently less perfectly burned, and 




up-str£am_e>id. down-stream end. down-stream end. three pipes. 
Fig. 98.— Standard Vitrified-pipe Culvert. Plant System. (1891.) 

the supposed extra strength is not therefore obtained. In Fig. 
98 are shown the standard plans for vitrified- pipe culverts as used 
on the "Plant system." Tile pipe is much clieaper than iron 
pipe, but is made in much shorter lengths and requires nmch 
more work in laying and especially to obtain a uniform grade. 



212 



RAILROAD CONSTRUCTION. 



§188. 



BOX CULVERTS. 



188. Wooden box culverts. This form serves the purpose of 
a cheap temporary construction which allows the use of a bal- 
lasted roadbed. As in all temporary constructions, the area 
should be made considerably larger than the calculated area 
(§§ 179-182), not only for safety but also in order that, if the 
smaller area is demonstrated to be sufficiently large, the per- 
manent construction (probably pipe) may be placed inside with- 
out disturbing the embankment. All designs agree in using 
heavy timbers (12'' X 12", 10'' X 12", or 8" X 12") for the 
side walls, cross-timbers for the roof, every fifth or sixth timber 
being notched down so as to take up the thrust of the side walls, 
and planks for the flooring. Fig. 99 shows some of the standard 
designs as used by the C, M. tfe St. P. Ry. 




Fig. 99.— Standard Timber Box Culvert. C, M. & St. P. Ry. (Feb. 1889. > 

189. Stone box culverts. In localities where a good quality 
of stone is cheap, stone box culverts are the cheapest form of 
permanent construction for culverts of medium capacity, but 
their use is decreasing owing to the frequent difficulty in obtain- 
ing really suitable stone within a reasonable distance of the 
culvert. The clear span of the cover-stones varies from 2 to 4 
feet. The required thickness of the cover-stones is sometimes 
calculated by the theory of transverse strains on the basis of cer- 
tain assumptions of loading — as a function of the height of the 
embankment and the unit strength of the stone used. Such a 
method is simply another illustration of a class of calculations 



§ 190. CULVERTS AND MINOR BRIDGES. 213 

wliicli look very precise and beautiful, but which are worse than 
useless (because misleading) on account of the hopeless uncertainty 
as to the true value of certain quantities which must be used in 
the computations. In the first place the true value of the unit 
tensile strength of stone is such an uncertain and variable 
quantity that calculations based on any assumed value for it are 
of small reliability. In the second place the weight of the prism 
of earth lying directly above the stone, plus an allowance for live 
load, is by no means a measure of the load on the stone nor of 
the forces that. tend to fracture it. All earthwork will tend to 
form an arch above any cavity and thus relieve an imcertain and 
probably variable proportion of the pressure that might other- 
wise exist. The higher the embankment the less the propor- 
tionate loading, until at some uncertain height an increase in 
height will not increase the load on the cover-stones. The effect 
of frost is likewise large, but uncertain and not computable. The 
usual practice is therefore to make the thickness such as experi- 
ence has shown to be safe with a good quality of stone, i.e., 
about 10 or 12 inches for 2 feet span and up to 16 or 18 inches 
for 4 feet span. The side walls should be carried down deep 
enough to prevent their being undermined by scour or heaved 
by frost. The use of cement mortar is also an important feature 
of first-class work, especially when there is a rapid scouring cur- 
rent or a liability that the culvert will run under a head. In 
Fig. 100 are shown standard plans for single and double stone box 
culverts as used on the IS^orfolk and Western R.R. 

190. Old-rail culverts. It sometimes happens (although very 
rarely) that it is necessary to bring the grade line within 3 or 4 
feet of the bottom of a stream and yet allosv an area of 10 or 12 
square feet. A single large pipe of sufficient area could not be 
used in this case. The use of several smaller pipes side by side 
would be both expensive and inefficient. For similar reasons 
neither wooden nor stone box culverts could be used. In sucli 
cases, as well as in many others where the head-room is not so 
limited, the plan illustrated in Fig. 101 is a very satisfactory 
solution of the problem. The old rails, having a length of 8 or 



214 



RAILROAD CONSTRUCTION. 



§ 190. 




191. 



CULVERTS AND MINOR BRIDGES. 



215 



9 feet, are laid close togetlier across a G-foot opeiiing. Some- 
times the rails are held together by long bolts passing through 
the webs of the rails. In the plan shown the rails are confined 



[TnTTTT TTTTTTr TTr.TTTT,TT;TT TTTrTTTTTTTrTr 7?1 




^fe 






Fig. 101.— Standard Old-rail Culvert. N. & W. R.R. (1895.) 

by low end walls on each abutment. This plan requires only 15 
inches between the base of the rail and the top of the culvert 
channel. It also gives a continuous ballasted roadbed. 



ARCH CULVERTS. 

191. Influence of design on flow. The variations in the 
design of arch culverts have a very marked influence on the 
cost and efficiency. To combine the least cost with the great- 
est efficiency, due weight should be given to the following 
elements : (a) the amount of masonry, (h) the simplicity of 
the constructive work, {c) the design of the wing walls, {d ) 
the design of the junction of the wing walls with the barrel 



LI 

(a) 




Fio. 102. — Types of Culverts. 

and faces of the arch, and {e) the safety and permanency of the 
construction. These elements are more or less antagonistic to 
each other, and the defects of most designs are due to a lack of 
proper proportion in the design of these opposing interests. The 
simplest construction (satisfying elements 1> and e) is the straight 



216 RAILROAD CONSTRUCTION. § 192. 

barrel arch between two parallel vertical head walls, as sketched 
in Fig. 102, a. From a hydraulic standpoint the design is poor, 
as the water eddies around the corners, causing a great resistance 
which decreases the flow. Fig. 102, 5, shows a much better de- 
sign in many respects, but much depends on the details of the 
design as indicated in elements {h) and (c/). As a general thing 
a good hydraulic design requires complicated and expensive 
masonry construction, i.e., elements (b) and {d) are opposed. 
Design 102, (?, is sometimes inapplicable because the water is 
liable to work in behind the masonry during floods and perhaps 
cause scour. This design uses less masonry than {a) or (h). 

192. Example of arch culvert design. In Plate XY is shown 
the design for an 8-foot arch culvert according to the standard 
of the ISTorfolk and Western R.R. Note that the plan uses 
the flaring wing walls (Fig. 102, h) on the up-stream side 
(thus protecting the abutments from scour) and straight wing 
walls (similar to Fig. 102, c) on the down-stream end. This 
economizes masonry and also simplifies the constructive work. 
Note also the simplicity of the junction of the wing walls with 
the barrel of the arch, there being no re-entrant angles below 
the springing line of the arch. The design here shown is but 
one of a set of designs for arches varying in span from 6' to 30'. 

MINOR OPENINGS. 

193. Cattle-guards, (a) Pit guards. Cattle-guards will be 
considered under the head of minor openings, since the old- 
fashioned plan of pit guards, which are even now defended and 
preferred by some railroad men, requires a break in the con- 
tinuity of the roadbed. A pit about three feet deep, five feet 
long, and as wide as the width of the roadbed, is w^alled up with 
stone (sometimes with wood), and the rails are supported on heavy 
timbers laid longitudinally with the rails. The break in the 
continuity of the roadbed produces a disturbance in the elastic 
wave running through the I'ails, the effect of which is noticeable 
at high velocities. The greatest objection, however, lies in the 



--41^2-- 




> 

X 



§ 193. 



CULVERTS AND MINOR BRIDGES. 



217 



dangerous consequences of a derailment or a failure of the tim- 
bers owing to unobserved decay or destruction by fire — caused 
perhaps by sparks and cinders from passing locomotives. The 
very insignificance of the structure often leads to careless in- 



'■■! .t i:; o" 1 



12 X Vl\ l.i 8' 




m^. 



12 10- 






-Cil2' 



Fig. 103.— Pit Cattle-guahds. P. R.R. 

spection. But if a single pair of wheels gets off the rails and 
drops into the pit, a costly wreck is inevitable. The (once) 
standard design for such a structure on the Pennsylvania E.R. 
is shown in Fig. 103. 

(b) Surface cattle-guards. These are fastened on top of the 
ties; the continuity of the roadbed is absolutely unbroken and 
thus is avoided much of the danger of a bad wreck owing to a 
possible derailment. The device consists essentially of overlay- 
ing the ties (both inside and outside the rails) with a surface on 
which cattle w411 not walk. The multitudinous designs for such 
a surface are variously effective in this respect. An objection, 
which is often urged indiscriminately against all such designs, is 
the liability that a brake-chain which may happen to be drag- 
ging may catch in the rough bars which are used. The bars 
are sometimes "home-made," of wood, as shown in Fig. 104. 
Iron, or steel bars are made as shown in Fig. 105. The 
general construction is the same as for the wooden bars. The 



218 



MAILROAD CONSTRUCTION. 



194. 



metal bars have far greater dnrabilitv, and it is claimed that they 
are more effective in discouraging cattle from attempting to 
cross. 




Fig. 104.— Cattle-guard with Wooden Slats. 




Fig. 105.— Merrill- Stevexs Steel Cattle-guard. 



194. Cattle-passes. Frequently when a railroad crosses a 
farm on an embankment, cutting the farm into two parts, the 
railroad company is obliged to agree to make a passageway 
through the embankment sufficient for the passage of cattle and 
perhaps even farm-was^ons. If the embankment is hi^h enousrh 
so that a stone arch is practicable, the initial cost is the only 
great objection to such a construction ; but if an open wooden 
structure is necessary, all the objections against the old-fashioned 
cattle-guarda apply witli equal force here. The avoidance of a 
grade crossing which would otherwise be necessary is one of the 



PLATE XVL 



iH BOLT EVERY THIRD TIE. 
-N,, O- 








STANDARD I-BRIDGES-14-FT. SPAN. 

NORFOLK AND WESTERN R.R. 
(1891.) 





TYPES OF PLATE GIRDER BRIDGES. 

C. M. & St.P. RY. 
(Dec. 1895.) 



TYPE '•£" GIRDER 

25 FEET AND UNDER, 




(To face page 219.) 



§ 195. CULVERTS AND MINOR BHTDGE8. 219 

• 

great compensations for the expense of the construction and 
maintenance of these structures. The construction is sometimes 
made by placing two pile trestle bents about 6 to 8 feet apart, 
supporting the rails by stringers in the usual way, the special 
feature of this construction being that the embankments are 
filled in behind the trestle bents, and the thrust of the embank- 
ments is mutually taken up through the stringers, which are 
notched at the ends or otherwise constructed so that they may 
take up such a thrust. The designs for old-rail culverts and 
arch culverts are also utilized for cattle-passes when suitable and 
convenient, as well as the designs illustrated in the following 
section. 

195. Standard stringer and I-beam bridges. The advantages 
of standard designs apply even to the covering of short spans 
with wooden stringers or with I beams — especially since 
the methods do not require much vertical space between the 
rails and the upper side of the clear opening, a feature which is 
often of prime importance. These designs are chiefly used for 
culverts or cattle-passes and for crossing over highways — pro- 
viding such a narrow opening would be tolerated. The plans 
all imply stone abutments, or at least abutments of sufficient 
stability to withstand all thrust of the embankments. Some of 
the designs are illustrated in Plate XVI. The preparation of 
these standard desii^ns should be attacked bv the same oreneral 
methods as already illustrated in § 156. When computing the 
required transverse strength, due allowance should be made for 
lateral bracing, which should be amply provided for. Xote 
particularly the methods of bracing illustrated in Plate XYI. 
The designs calling for iron (or steel) stringers may be classed 
as permanent constructions, which are cheap, sate, easily in- 
spected and maintained and therefore a desirable method of 
construction. 



CHAPTER YII. 
BALLAST. 

196. Purpose and requirements. " The object of the ballast 
is to transfer the applied load over a large surface ; to hold the 
timber work in place horizontally ; to carry off the rain-water 
from the superstructure and to prevent freezing up in winter ; 
to afford means of keeping the ties truly up to the grade line ; 
and to give elasticity to the roadbed." This extremely con- 
densed statement is a description of an ideally perfect ballast. 
The value of any given kind of ballast is proportional to the 
extent to which it fulfills these requirements. The ideally per- 
fect ballast is not necessarily the most economical ballast for all 
roads. Light traffic generally justifies something cheaper, but 
a very common error is to use a very cheap ballast when a small 
additional expenditure would procure a much better ballast 
which would be much more economical in the long run. 

197. Materials. The materials most commonly employed are 
gravel and broken stone. Burnt clay, cinders, shells, and small 
coal are occasionally used as ballast when they are especially 
cheap and convenient or when better kinds are especially expen- 
sive. Although it is hardly correct to speak of the natural soil 
as ballast, yet many miles of cheap railways are "ballasted" 
with the natural soil, which is then called " mud ballast." 

Mud ballast. When the natural soil is gravelly so that rain 
will drain through it quickly, it will make a fair roadbed for 
light traffic, but for heavy traffic, and for the greater part of the 
length of most roads, the natural soil is a very poor material for 
ballast ; for, no matter how suitable the soil might be along 

220 



§ 197. BALLAST. 221 

limited sections of the road, it would practically never liappen 
that the soil would be uniformly good throughout the whole 
length of the road. Considering that a heavy rain will in one 
day spoil the results of weeks of patient " surfacing" with mud 
ballast, it is seldom economical to use "mud" if there is a 
gravel-bed or other source of ballast anywhere on the line of 
the road. 

Cinders. Tlie advantages consist in the excellent facilities 
for drainage, ease of handling, and cheapness — after the road is 
in operation. One disadvantage is excessive dust in dry weather. 
Cinders are considered preferable to gravel in yards. 

Slag. When slag is readily obtainable it furnishes an ex- 
cellent ballast, free from dust and perfect in drainage qualities. 
Some kinds of slag are objectionable on account of their delete- 
rious chemical effect on the ties and spikes — especially on 
metallic ties. 

Shells, small coal, etc. These comparatively inferior kinds 
of ballast are used for light traffic when they are especially cheap 
and convenient. They are extremely dusty in dry weather, 
break up into very fine dust, and are but little better than mud. 

Gravel. This is the most common form of ballast which 
may be called good ballast. In 1885, the Roadmasters Associa- 
tion of America voted in favor of gravel ballast as against rock 
ballast. Although not so stated, this action was perhaps due to 
a conviction of its real economy for the average railroad of this 
country, which may be called a "light traffic" road. Gravel 
should preferably be screened over a screen having a \" mesh, 
so as to screen out all dirt and the finest stones. Generally a 
railroad will be able to find at some point along its line a 
"gravel-pit" affording a suitable supply. This may be dug out 
with a steam-shovel, screened if necessary, and sent out over 
the line by the train-load at a comparatively small cost. 

Rock or broken stone. Ruck ballast is generally specified to 
be such as will pass through a \\" (or 2'') ring. Although pref- 
erably broken by hand, machine-broken stone is much cheaper. 
It is most easily handled with forks. This also has the effect of 



222 RAILROAD CONSTRUCTION. § 198. 

screening out tlie dirt and fine chips which would interfere with 
eliectual drainage. Rock ballast is more expensive in first cost, 
and also more troublesome to handle, than any other kind, but 
under heavy trafiic will keep in surface better and will require 
less work for maintenance after the ties have become thoroughly 
bedded. For roads with very light traffic, running few trains, 
at comparatively low velocities, the advantages of rock ballast 
over other kinds are not so pronounced. For such roads rock 
ballast is an expensive luxury. The amount of trafi^ic which 
will justify the use of rock ballast will depend on the cost of 
obtaining ballast of the various kinds. 

198. Cross-sections. A depth of 12'' under the tie is gener- 
ally required on the best roads, but for light trafiic this is some- 
times reduced to Q" and even less. The width is generally 1 to 
2 feet less than the wddth of the roadbed proper — excluding 
ditches. If the ballast has an average width of 10 feet (12 feet 
at bottom and 8 feet at top) and an average depth of 15 inches 
(including that placed between the ties), it will require 2144 
cubic yards per mile of track. The P. R.R. estimates 2500 
cubic yards of gravel and 2800 cubic yards of stone ballast per 
mile of single track. On account of the requirements of drain- 
age the best form of cross-section depends on the kind of ballast 
used. 

Mud ballast. Since the great objection to mud ballast lies in 
its liability to become soft by soaking up the rain that falls, it 
becomes necessary that it should be drained as quickly and 
readily as its nature will permit. Fig. 106 shows a typical 



Fig. OP.— '• Mud " Ballast. 

cross-section for mud ballast. It should be crowned 2" above 
the top of the tie at the center, thence sloped so as to leave a 
slight clearance under the rail between the ties, thence sloping 
down to the bottom of the tie at each end and continuing to 



§199. 



BALLAST. 



223 



slope down to the ditcli (in cut), wliicli should be 18" or 20" be- 
low the bottom of the tie. 

Gravel, cinders, slag, etc. The subgrade is crowned 6" or 
8" in the center, as shown in Fig. 107. The ballast is crowned 




Fig. 107.— Gravel Ballast. 

to the top of the tie in the center, but is sloped down to the 
bottom of the tie at each end. This is necessary (and more 
especially so with mud ballast) to prevent a possible accumula- 
tion and settlement of water at the ends of the tie, which would 
readily soak into the end fibers and produce decay. 

Broken stone. Stone ballast is shouldered out beyond the 
ends of the ties so as to aiford greater lateral binding. The 
space betAveen the ties is filled up level with the tops. The 




Fig. 108.— Broken Stone Bali^ast. 

perfect drainage of stone ballast permits this to be done w^ithout 
any danger of causing decay of the ties by the accumulation and 
retention of water. 

199. Methods of laying ballast. The cheapest method of 
laying ballast on new roads is to lay ties and rails directly on 
the prepared subgrade and run a construction train over the 
track to distribute the ballast. Then the track is lifted up until 
sufficient ballast is worked under the ties and the track is prop- 
erly surfaced. This method, although cheap, is apt to injure 
the rails by causing bends and kinks, due to the passage of 
loaded construction trains when the ties are very unevenly and 
roughly supported, and the method is therefore condemned and 
prohibited in some specifications. The best method is to draw 



224 RAILROAD CONSTRUCTION, § 200. 

ill carts (or on a contractor's temporary track) the ballast that is 
required under the level of the hottoin of the ties. Spread this 
ballast carefully to the required surface. Then lay the ties and 
rails, which will then have a very fair surface and uniform sup- 
port. A construction train can then be run on the rails and 
distribute sufficient additional ballast to pack around and between 
the ties and make the required cross-section. 

The necessity for constructing some lines at an absolute 
minimum of cost and of opening them for traffic as soon as pos- 
sible has often led to the policy of starting traffic when there is 
little or no ballast — perhaps nothing more than a mere tamping 
of the natural soil under the ties. When this is done ballast 
may subsequently be drawn where required by the train-load on 
flat cars and unloaded at a minimum of cost by means of a 
'' plough." The plough has the same width as the cars and is 
guided either by a ridge along the center of each car or by short 
posts set up at the sides of the cars. It is drawn from one end 
of the train to the other by means of a cable. The cable is 
sometimes operated by means of a small hoisting-engine carried 
on a car at one end of the train. Sometimes the locomotive is 
detached temporarily from the train and is run ahead with the 
cable attached to it. 

200. Cost. The cost of ballast in the trade is quite a variable 
item for different roads, since it depends (a) on the first cost of 
the material as it comes to the road, (h) on the distance from 
the source of supply to the place where it is used, and {c) on 
the method of handling. The first cost of cinder or slag is 
frequently insignificant. A gravel-pit may cost nothing except 
the price of a little additional land beyond the usual limits of the 
right of way. Broken stone will usually cost $1 or more per 
cubic yard. If suitable stone is obtainable on the company's 
land, the cost of blasting and breaking should be somewhat less 
than this. The cost of loading the ballast on to trains will be 
small (per cubic yard) if it is handled with steam-shovels — as in 
the case of gravel taken from a gravehpit. Hand-shovelling 
will cost more. The cost of hauling will depend on the distance 



§ 200 BALLAST, 225 

hauled, and also, to a considerable extent, on the limitations on 
the operation of the train due to the necessity of keeping out of 
the way of regular trains. There is often a needless waste in 
this way. The ''mud train " is considered a pariah and entitled 
to no rights whatever, regardless of the large daily cost of such 
a train and of the necessary gang of men. The cost of broken 
stone ballast m the track is estimated at $1.25 per cubic yard. 
The cost of gravel ballast is estimated at 60 c. per cubic yard 
in the track. The cost of placing and tamping gravel ballast is 
estimated at 20 c. to 24 c. per cubic yard, for cinders 12 c. to 
15 c. per cubic yard. The cost of loading gravel on cars, usintr 
a steam-shovel, is estimated at 6 c. to 10 c. per cubic yard.^ 

* Report Roadmasters Association, 1885. 



CHAPTER YIII. 
TIES, 

AND OTHER FORMS OF RAIL SUPPORT. 

201. Various methods of supporting rails. It is necessary 
that the rails shall be sufficiently supported and braced, so that 
the gauge shall be kept constant and that the rails shall not be 
subjected to excessive transverse stress. It is also preferable 
that the rail support shall be neither rigid (as if on solid rock) 
nor too yielding, but shall have a uniform elasticity throughout. 
These requirements are more or less fulfilled by the following 
methods. 

(a) Longitudinals. Supporting the rails throughout their 
entire length. This method is very seldom used in this country 
except occasionally on bridges and in terminals when the 
longitudinals are supported on cross- ties. In § 224: will be 
described a system of rails, used to some extent in Europe, 
having such broad bases that they are self-supporting on the 
ballast and are only connected by tie-rods to maintain the 
gauge. 

(b) Cast-iron "bowls" or "pots.'' These are castings resem- 
bling large inverted bowls or pots, having suitable chairs on 
top for holding and supporting the rails, and tied together 
with tie-rods. They will be described more fully later (§ 223). 

(b) Cross-ties of metal or wood. These will be discussed in 
the following sections. 

202. Economics of ties. The true cost of ties depends on the 
relative total cost of maintenance for long periods of time. The 
first cost of the ties delivered to the road is but one item in the 

226 



§ 203. TIES. 227 

economics of tlie question. Clieap ties require fre(|uent renew- 
als, whicli cost for the lahor of each renewal practically the 
same whether the tie is of oak or hemlock. Clieap ties make a 
poor roadbed which will require more track labor to keep even 
in tolerable condition. The roadbed will require to be disturbed 
so frequently on account of renewals that the ties never get an 
opportunity to get settled and to form a smooth roadbed for any 
length of time. Irregularity in width, thickness, or length of 
ties is especially detrimental in causing the ballast to act 
and wear unevenly. The life of ties has thus a more or less 
direct influence on the life of the rails, on the wear of rollinof 
stock, and on the speed of trains. _/ These last items are not so 
readily reducible to dollars and cents, but when it can be shown 
that the total cost, for a long period of time, of several renewals 
of cheap ties, with all the extra track labor involved, is as great as 
or greater than that of a few renewals of durable ties, then there 
is no question as to the real economy. In the following dis- 
cussions of the mei-its of untreated ties (either cheap or costly), 
chemically treated ties, or metal ties, the true question is there- 
fore of the ultimate cost of maintaining any particular kind of 
ties for an indefinite period, the cost including the flrst cost of 
the ties, the labor of placing them and maintaining them to 
surface, and the somewhat uncertain (but not therefore non- 
existent) effect of frequent renewals on repairs of rolling stock, 
on possible speed, etc. 

WOODEN TIES. 

203. Choice of wood. This naturally depends, for any partic- 
ular section of country, on the supply of wood wliicli is most 
readily available. The woods most commonly used, especially 
in this country, are oak and pine, oak being the most durable 
and generally the most expensive. Kedwood is used very ex- 
tensively in California and proves to be extremely durable, so 
far as decay is concerned, but it is very soft and is much injured 
by " rail-cutting." This defect is being partly remedied by the 



228 RAILROAD CONSTRUCTION. % 204. 

use of tie-plates, as will be explained later. Cedar, chestnut, 
liemlock, and tamarack are frequently used in this country, . In 
tropical countries very durable ties are frequently obtained from 
the hard woods peculiar to those countries. According to a re- 
cent bulletin of tlie U. S. Department of Agriculture the pro- 
portions of the various kinds used in the United States are about 
as follows : 



Oak 60^ 

Pine 20 

Cedar 6 



Chestnut 5j 

Hemlock and Tama- 
rack 3 

Redwood 3 



Cypress 2% 

Various 1 



Total 100^ 



204. Durability. The durability of ties depends on the cli- 
mate ; the drainage of the ballast ; the volume, weight, and 
speed of the traffic ; the curvature, if any ; the use of tie-plates ; 
the time of year of cutting the timber ; the age of the timber 
and the degree of its seasoning before placing in the track ; the 
nature of the soil in which the timber was grown; and, chiefly, 
on the species of wood employed. The variability in these 
items will account for the discrepancies in the reports on the life 
of various woods used for ties. 

White oak is credited with a life of 5 to 12 years, depending 
principally on the traffic. Is is both hard and durable, the 
hardness enabling it to withstand the cutting tendency of the 
rail-flanges, and the durability enabling it to resist decay. Pine 
and redwood resist decay very well, but are so soft that they are 
badly cut by the rail-flanges and do not hold the spikes very 
well, necessitating frequent respiking. Since the spikes must 
be driven within certain very limited areas on the face of each 
tie, it does not require many spike-holes to '^ spike-kill " the 
tie. On sharp curves, especially with heavy traffic, the vdieel- 
flange pressure produces a side pressure on the rail tending to 
overturn it, which tendency is resisted by the spike, aided some- 
times by rail-braces. Whenever the pressure becomes too great 
the spike will yield somewhat and will be slightly withdrawn. 
The resistance is then somewhat less and the spike is soon so loose 
that it must be redriven in a new hole. If this occurs very 



§206. TIES. 229 

often, the tie may need to be replaced long before any decay has 
set in. When the traffic is v^ery light, the wood very durable, 
and the climate favorable ties have been known to last 25 years. 

205. Dimensions. The usual dimensions for the best roads 
(standard gauge) are 8' to 8' <6" long, 6" to 7" thick, and 8" to 
10" wide on top and bottom (if they are hewed) or 8" to 9" 
wide if they are sawed. For cheap roads and light traffic the 
length is shortened sometimes to 7' and the cross-section also re- 
duced. On the other hand a very few roads use ties 9' long. 

Two objections are urged against sawed ties : first, that the 
grain is torn by the saw, leaving a woolly surface which induces 
decay ; and secondly, that, since timber is not perfectly straight- 
grained, some of the fibers are cut obliquely, exposing their ends, 
which are thus liable to decay. The use of a " planer-saw " ob- 
viates the first difficulty. Chemical treatment of ties obviates 
both of these difficulties. Sawed ties are more convenient to 
handle, are a necessity on bridges and trestles, and it is even, 
clahned, although against connnonly received opinion, that 
actual trial has demonstrated that they are more durable than 
hewed ties. 

206. Spacing. The spacing is usually 14 to 16 ties to a 30- 
foot rail. This number is sometimes reduced to 12 and even 
10, and on the other hand occasionally increased to 18 or 20 by 
employing narrower ties. There is no economy in reducing the 
number of ties very nuich, since for any required stiffness of 
track it is more economical to increase the number of supports than 
to increase the weight of the rail. The decreasinir cost of rails 
and the increasing cost of ties have materially changed the rela- 
tion between number of ties and weight of rail to produce a 
given stiffness at minimum cost, but many roads have found it 
economical to employ a large number of ties rather than increase 
the weight of the rail. On the other hand there is a practical 
limit to the number that may be employed, on account of the 
necessary space between the ties that is required for proper 
tamping. This width is ordinarily about twice the width of the 
tie. At this rate, with light ties 6" wide and with 12^' clear 



230 RAILROAD CONSTRUCTION. § 207. 

space, there would be 20 ties per 30-foot rail, or 3520 per mile. 
The smaller ties can generally be bought much cheaper (propor- 
tionately) than the larger sizes, and hence the economy. 

Track instructions to foremen generally require that the 
spacing of ties shall 7iot be uniform along the lengtli of any 
rail. Since the joint is generally the weakest part of the rail 
structure, the joint requires more support than the center of the 
rail. Therefore the ties are placed with but 8" or 10'' clear 
space between them at the joints, this applying to 3 or 4 ties at 
each joint ; the remaining ties, required for each rail length, are 
equally spaced along the remaining distance. 

207. Specifications. The specifications for ties are apt to 
include the items of size, kind of wood, and method of con- 
struction, besides other minor directions about time of cutting, 
seasoning, delivery, quality of timber, etc. 

(a) Size. The particular size or sizes required will be some- 
what as indicated in § 205. 

(b) Kind of wood. When the kind or kinds of wood are spe- 
cified, the most suitable kinds that are available in that section 
of country are usually required. 

(c) Method of construction. It is generally specified that the 
ties shall be hewed on two sides; that the two faces thus made 
shall be parallel planes and that the bark shall be removed. It 
is sometimes required that the ends shall be sawed off square ; 
that the timber shall be cut in the w^inter (when the sap is down) ; 
and that the ties shall be seasoned for six months. These last 
specifications are not required or lived up to as much as their 
importance deserves. It is sometimes required that the ties shall 
be delivered on the right of way, neatly piled in rows, the alter- 
nate rows at right angles, piled if possible on ground not lower 
than the rails and at least seven feet away from them, the lower 
row of ties resting on two ties which are themselves supported 
so as to be clear of the ground. 

(d) Q,uality of timber. The usual specifications for sound 
timber are required, except that they are not so rigid as for a 
better class of timber work. The ties must be sound, reason- 



§ 208. TIES. 231 

ably straight-grained, and not very crooked — one test being that 
a hne joining the center of one end with the center of the middle 
shall not pass outside of the other end. Splits or shakes, espe- 
<ciallj if severe, should cause rejection. 

Specifications sometimes require that the ties shall be cut 
from single trees, making what is known as "pole ties" and 

definitely condemnino: those which ____„ _____ 

are cut or split from larger trunks, ( ^ ) 1 

giving two "slab ties" or four ,„,,,„. 3,,,,,,. quartert,. 

''quarter ties", for each cross- Fig. 109.— Methods of cutting 

section, as is illustrated in Fig. Ties. 

109. Even if pole ties are better, their exclusive use means the 

rapid destruction of forests of young trees. 

208. Regulations for laying and renewing ties. The regula- 
tions issued by railroad companies to their track foremen will 
generally include the following, in addition to directions regard- 
ing dimensions, spacing, and specifications given in §§ 20t1:-207. 
When hewn ties of somewhat variable size are used, as is fre- 
quently the case, the largest and best are to be selected for use 
as joint ties. If the upper surface of a tie is found to be warped 
(contrary to the usual specifications) so that one or both rails do 
not get a full bearing across the wdiole width of the tie, it must 
be adzed to a true surface along its whole length and not merely 
notched for a rail-seat. AYhen respiking is necessary and spikes 
have been pulled out, the holes should be immediately plutrged 
with "wooden spikes," which are supplied to the foremen for 
that express purpose, so as to fill up the holes and prevent the 
decay which would otherwise take place when the hole becomes 
filled wdth rain-water. Ties sliould always be laid at right angles 
to the rails and never obliquely. Minute regulations to prevent 
premature rejection and renewal of ties are frequently made. It 
is generally required that the requisitions for renewals shall be 
made by the actual count of the individual ties to be renewed 
instead of l)y any wholesale estimates. It is unwise to have ties 
of widely variable size, hardness, or durability adjacent to each 



232 BAILROAB CONSTRUCTION. § 209. 

other in the track, for the uniform elasticity, so necessary for 
smooth riding, will be unobtainable under those circumstances. 

209. Cost of ties. When railroads can obtain ties cut by 
farmers from woodlands in the immediate neighborhood, the 
price will frequently be as low as 20 c. for the smaller sizes, 
running up to 50 c. for the larger sizes and better qualities, espe- 
cially when the timber is not very plentiful. Sometimes if a 
railroad cannot procure suitable ties from its immediate neigh- 
borhood, it will find that adjacent railroads control all adjacent 
sources of supply for their own use and that ties can only be 
procured from a considerable distance, with a considerable added 
cost for transportation. First-class oak ties cost about 75 to 80 c. 
and frequently much more. Hemlock ties can generally be 
obtained for 35 c. or less. 

PKESERVATIVE PROCESSES FOR WOODEN TIES. 

210. General principle. Wood has a fibrous cellular struc- 
ture, the cells being filled with sap or air. The woody fiber is 
but little subject to decay unless the sap undergoes fermentation. 
Preservative processes generally aim at removing as much of the 
w^ater and sap as possible and filling up the pores of the wood 
with an antiseptic compound. The most common methods (ex- 
cept one) all agree in this general process and only differ in the 
method employed to get rid of the sap and in the antiseptic 
chemical with which the fibers are filled. One valuable feature 
of these processes lies in the fact that the softer cheaper woods 
(such as hemlock and pine) are more readily treated than are the 
harder woods and yet will produce practically as good a tie as a 
treated hard-wood tie and a very much better tie than an un- 
treated hard -wood tie. The various processes will be briefly 
described, taking up first the process which is fundamentally 
different from the others, viz., vulcanizing. 

211. Vulcanizing. The process consists in heating the timber 
to a temperature of 300° to 500° F. in a cylinder, the air being 
under a pressure of 100 to 175 lbs. per square inch. By this 
process the albumen in the sap is coagulated, the water evap- 



§ 21-2. TIES. 233 

orated, and the pores are partially closed by the coagulation of 
the albumen. It is claimed that the heat sterilizes the wood and 
produces chemical changes in the wood which give it an antisep- 
tic character. It has been very extensively used on the elevated 
lines of New York City, and it is claimed to give perfect satis- 
faction. The treatment has cost that road 25 c. ])cr tie. 

212. Creosoting. This process consists in impregnating the 
wood with toood-creosote or with dead oil of coal-tar. Wood- 
creosote is one of the products of the destructive distillation of 
wood — usually long-leaf pine. Dead oil of cocd-tar is a prod- 
uct of the distillation of coal-tar at a temperature between 480° 
and 760° F. It would require about 35 to 50 pounds of creo- 
sote to completely till the pores of a cubic foot of w^ood. But 
it would be impossible to force such an amount into the wood, 
nor is it necessary or desirable. About 10 pounds per cubic 
foot, or about 35 pounds per tie, is all that is necessary. For 
piling placed in salt water about 18 to 20 pounds per cubic foot 
is used, and the timber is then perfectly protected against the 
ravages of the teredo navcdis. To do the work, long cylinders, 
which may be opened at the ends, are necessary. Usually the 
timbers are run in and out on iron carriaii:es runninof on rails 
fastened to braces on the inside of the cylinder. AVhen the load 
has been run in, the ends of the cylinder are fastened on. The 
water and air in the pores of the wood are first drawn out by 
subjecting the wood alternately to steam-pressure and to the 
action of a vacuum-pump. This is continued for several hours. 
Then, after one of the vacuum 23eriods, the cylinder is filled 
with creosote oil at a temperature of about 170° F. The pumps 
are kept at work until the pressure is about 80 to 100 pounds 
per square inch, and is maintained at this pressure from one to 
two hours according to the size of the timber. The oil is then 
withdrawn, the cylinders opened, the train pulled out and an- 
other load made up in 40 to 60 minutes. The average time re- 
quired for treating a load is about 18 or 20 hours, the absorption 
about 10 or 11 pounds of oil per cubic foot, and the cost (1894) 
from $12.50 to 814.50 per thousand feet B. M. 



234 RAILROAD CONSTRUCTION. §213. 

213. Burnettizing (chloride -of -zinc process). This process is 
very similar to the creosotiiig process except that the chemical is 
chloride of zinc, and that tlie chemical is not heated before use. 
The preliminary treatment of the wood to alternate vacuum and 
pressure is not continued for quite so long a period as in the 
creosoting process. Care must be taken, in using this process, 
that the ties are of as uniform quality as possible, for seasoned 
ties will absorb much more zinc chloride than unseasoned (in the 
same time), and the product will lack uniformity unless the sea- 
soning is uniform. The A., T. & S. Fe K.R. has works of its 
own at which ties are treated by this process at a cost of about 
25 c. per tie. The Southern Pacific R.R. also has works for 
burnettizing ties at a cost of 9.5 to 12 c. per tie. The zinc- 
chloride solution used in these works contains only 1.7^ of zinc 
chloride instead of over 3^ as used in the Santa Fe works, which 
perhaps accounts partially for the great difference in cost per tie. 
One great objection to burnettized ties is the fact that the chem- 
ical is somewhat easily washed out, when the wood again be- 
comes subject to decay. Another objection, which is more 
forcible with respect to timber subject to great stresses, as in 
trestles, than to ties, is the fact that when the solution of zinc 
chloride is made strong (over 3^) the timber is made very brittle 
and its strength is reduced. The reduction in strength has been 
shown by tests to amount to J to -^-^ of the ultimate strength, 
and that the elastic limit has been reduced by about \. 

214. Kyanizing (bichloride-of-mercury or corrosive-sublimate 
process) . This is a process of ' ' steeping. ' ' It requires a much 
longer time than the previously described processes, but does not 
require such an expensive plant. Wooden tanks of sufficient 
size for the timber are all that is necessary. The corrosive subli- 
mate is first made into a concentrated solution of one part of 
chemical to six parts of hot water. When used in the tanks this 
solution is weakened to 1 part in 100 or 150. The wood will 
absorb about 5 to 6.5 pounds of the bichloride per 100 cubic 
feet, or about one pound for each 4 to ties. The timber is 
allowed to soak in the tanks for several days, the general rule 



§ 215. TIES. 235 

being about one day for each inch of least thickness and one day 
over — wliich means seven days for six-inch ties, or thirteen (to 
fifteen) days for 1'2" timber (least dimension). The process is 
somewhat objectionable on account of the chemical being such a 
virulent poison, workmen sometimes being sickened by the fumes 
arising from the tanks. On the Baden railway (Germany) 
kyanized ties last 20 to 30 years. On this railway the wood is 
always air-dried for two weeks after impregnation and before 
being used, which is thought to have an important effect on its 
durability. The solubility of the chemical and the liability of 
the chemical washing out and leaving the wood unprotected is 
an element of weakness in the method. 

215. Wellhouse (or zinc-tannin) process. The last two 
methods described (as well as some others employing siinilar 
chemicals) are open to the objection that since the wood is im- 
pregnated with an aqueous solution, it is liable to be washed out 
very rapidly if the wood is placed under water, and will even 
disappear, although more slowly, under the action of moisture 
and rain. Several processes have been proposed or patented to 
prevent this. Many of them belong to one class, of which the 
Wellhouse process is a sample. By these processes the timber 
is successively subjected to the action of two chemicals, each 
individually soluble in water, and hence readily impregnating 
the timber, but the chemicals when brought in contact form in- 
soluble compounds which cannot be washed out of the w^ood- 
cells. By the Wellhouse process, the wood is first impregnated 
with a solution of chloride of zinc and glue, and is then subjected 
to a bath of tannin under pressure. The glue and tannin com- 
bine to form an insoluble leathery compound in the cells, wliich 
will prevent the zinc chloride from being washed out. It is 
being used by the A., T. <k S. Fe E,.R., their works being 
located at Las Yegas, New Mexico, and also by the Union 
Pacific E.B. at their works at Laramie, Wyo. In 1897 Mr. ,] . 
M. Meade, a resident engineer on the A., T. tfe S. Fe, exhibited 
to the Road masters Association of America a piece of a tie treated 
by this process which had been taken from the tracks after 



236 RAILROAD CONSTRUCTION. § 216. 

nearly 13 years' service. Tlie tie was selected at random, was 
taken out for the sole purpose of having a specimen, and Avas 
stiii in sound condition and capable of serving many years longer. 
The cost of the treatment was then quoted as 13 c. per tie. 
It was claimed that the treatment trebled the life of the tie 
besides adding to its spike-holding power. 

216. Cost of treating. The cost of treating ties by the vari- 
ous methods has been estimated as follows * — assuming that 
the plant was of sufficient papacity to do the work economi- 
cally : creosoting, 25 c. per tie; vulcanizing, 25 c. per tie; 
burnettizing (chloride of zinc), 8.25 c. per tie ; kyanizing 
(steeping in corrosive sublimate), 14.6 c. per tie; Wellhouse 
process (chloride of zinc and tannin), 11.25 c. per tie. These 
estimates are only for the net cost at the works and do not 
include the cost of hauling the ties to and from the works, which 
may mean 5 to 10 c. per tie. Some of these processes have 
been installed on cars which are transported over the road and 
operated where most convenient. 

217. Economics of treated ties. The fact that treated ties are 
not universally adopted is due to the argument that the added 
life of the tie is not worth the extra cost. If ties can be bought 
for 25 c, and cost 25 c. for treatment, and the treatment 
onlv doubles their life, there is apparently but little gained 
except the work of placing the extra tie in the track, which is 
more or less offset by the interest on 25 c. for the life of the 
untreated tie, and the larger initial outlay makes a stronger im- 
pression on the mind than the computed ultimate economy.' 
But when ties cost 75 c. and treatment costs only 25 c, 
or perhaps less, then the economy is more apparent and un- 
questionable. But this analysis may be made more closely. 
As shown in § 202, the disturbance of the roadbed on account 
of frequent renewals of untreated ties is a disadvantage which 
would justify an appreciable expenditure to avoid, although it is 



*Bull. No. 9, U. S. Dept. of Agric, Div. of Forestry. App. No. 1, hj 
Henry Flad. 



§ 217. TIKS. 237 

very difficult to closely estiuicate its true value. The annual cost 
of a system of ties may be considered as the sum of {a) the 
interest on the first cost, {h) the annual sinking fund that would 
buy a new tie at the end of its life, and (c) the average annual 
cost of maintenance for the life of the tie, which includes the 
cost of laying and the considerable amount of subsequent tamp- 
inoj that must be done until the tie is fairlv settled in the road- 
bed, beside the regular trackwork on the tie, which is practically 
constant. This last item is difficult to compute, but it is easy to 
see that, since the cost of laying the tie and the subsequent 
tamping to obtain proper settlement is the same for all ties (of 
similar form), the average annual charge on the longer-lived tie 
would be much less. In the following comparison item {e) is 
disregarded, simply remembering that the advantage is with the 
longer-lived tie. 



Untreated tie. 

Original cost 40 cents 

Life (assumed at) 7 years 



Treated tie. 

65 cents 
14 years 



Item {a) — interest on first cost @ 4^ 1.6 cents 

" [Jj) — sinking fund @ 4^ 5.1 " 

" (c') — (considered here as offsetted) . . . 



2.6 cents 
3.6 '' 



Average annual cost (except item {c)) .... 6.7 cents | 6.2 cents 

On this basis treated ties will cost 0.5 cent less per annum 
hesides the advantage of item (c) and the still more indefinite 
ad van tastes resultinor from smoother runnino^ of trains, less wear 
and tear on rolling stock, etc., due to less disturbance of the 
roadbed. 

In Europe, where wood is expensive, untreated ties are 
seldom used, as the treatment is always considered to be worth 
more than it costs. The rapid destruction of the forests of tim- 
ber in this country is having the effect of increasing the price, so 
that it will not be long before treated ties (or metal ties) will be 
economical for a large majority of the railroads of the country. 



22S RAILROAD CONSTRUCTION. § 218. 



METAL TIES. 



218. Extent of use. In 1894 ^ there were nearly 35000 miles 
of " metal track " in various parts of the world. Of this total, 
there were 3645 miles of "longitudinals " (see § 224), found ex- 
clusively in Europe, nearly all of it being in Germany. There 
were over 12000 miles of '" bowls and plates " (see § 223), found 
almost entirely in British India and in the Argentine Republic. 
The remainder, over 18000 miles, was laid with metal cross-ties 
of various designs. There were over 8000 miles of metal cross- 
ties in Germany alone, about 1500 miles in the rest of Europe 
over 6000 miles in British India, nearly 1000 miles in the rest 
of Asia, and about 1500 miles more in various other parts of the 
world. Several railroads in this country have tried various de- 
signs of these ties, but their use has never passed the experi- 
mental stage. These 35000 miles represent about 9^ of the 
total railroad mileage of the world — nearly 400000 miles. They 
represent about 17.6^ of the total railroad mileage, exclusive of 
the United States and Canada, where they are not used at all 
except experimentally. In the four years from 1890 to 1894 the 
use of metal track increased from less than 25000 miles to neai-ly 
35000 miles. This increase was practically equal to the total in- 
crease in railroad mileage during that time, exclusive of the in- 
crease in the United States and Canada. This indicates a laro-e 
growth in the percentage of metal track to total mileage, and 
therefore an increased appreciation of the advantages to be de- 
rived from their use. 

219. Durability. The durability of metal track is still far 
from being a settled question, due largely to the fact that the 
best form for such track is not yet determined, and that a large 
part of the apparent failures in metal track have been evidently 
due to defective design. Those in favor of them estimate the 
life as from 30 to 50 years. The opponents place it as not more 
than 20 years, or perhaps as long as the best of wooden ties. 

* Bulletin No. 9, U. S. Dept. of Agriculture, Div. of Forestry. 



§ 220. TIES. 239 

Unlike the wooden tie, however, whicli deteriorates as mueh 
with time as with usage, the life of a metal tie is more largely a 
function of the trafhc. The life of a well-designed metal tie has 
been estimated at 150000 to 200000 trains; for 20 trains per 
day, or say 6000 per year, this would mean from 25 to 33 years. 
20 trains per day on a single track is a much larger number than 
will be found on the majority of railroads. Metal ties are found 
to be subject to rust, especially when in damp localities, such as 
tunnels; but on the other hand it is in such confined localities, 
where renewals are troublesome, that it is especially desirable to 
employ the best and longest-liv^ed ties. Faint, tar, etc., have 
been tried as a protection against rust, but the efficacy of such 
protection is as yet uncertain, the conditions preventing any re- 
newal of the protection — such as may be done by repainting a 
bridge, for example. Failures in metal cross-ties have been 
largely due to cracks which begin at a corner of one of the square 
holes which are generally punched through the tie, the holes 
being. made for the bolts by whicli the rails are fastened to the 
tie. The holes are generally punched because it is cheaper. 
Heaming the holes after punching is thought to be a safeguard 
against this frequent cause of failure. Another method is to 
round the corners of the square punch with a radius of about 
1-'^ If a crack has already started, the spread of the crack may 
be prevented by drilling a small hole at the end of it. 

220. Form and dimensions of metal cross-ties. Since stability 
in the ballast is an essential quality for a tie, this must be accom- 
plished either by turning down the end of the tie or by having 
some form of lug extending downward from one or more points 
of the tie. The ties are sometimes depressed in the center (see 
Flate XYII, IS" Y. C. & H. R. R.R. tie) to allow for a thick 
covering of ballast on top in order to increase its stability in the 
ballast. This form requires that the ties should be sufficiently 
well tamped to prevent a tendency to bend out straight, thus 
widening the gauge. Many designs of ties are objection- 
able because they cannot be ])laced in the track without 
disturbing adjacent ties. The failure of many metal cross- 



240 RAILROAD CONSTRUCTION. § 221. 

ties, otherwise of good design, may be ascribed to too light 
weight. Those weighing nmch less than 100 pounds have 
proved too light. From 100 to 130 pounds weight is being used 
satisfactorily on German railroads. Tlie general outside dimen- 
sions are about the same as for wooden ties, except as to thick- 
ness. The metal is generally from J'' to f '' thick. They are, 
of course, only made of wrought iron or steel, cast iron being 
used only for "bowls " or " plates " (see § 213). The details 
of construction of some of the most commonly used ties may be 
seen by a study of Plate XVII. 

221. Fastenings. The devices for fastening the rails to the 
ties should be such that the gauge may be widened if desired on 
curves, also that the gauge can be made true regardless of slight 
inaccuracies in the manufacture of the ties, and also that shims 
may be placed under the rail if necessary during cold weather 
when the tie is frozen into the ballast and cannot be easily 
disturbed. Some methods of fastening require that the base of 
the rail be placed against a lug which is riveted to the tie or 
which forms a part of it. This has the advantage of reducing 
the number of pieces, but is apt to have one or more of the 
disadvantao^es named above. Metal kevs or wooden wedges are 

O tJ CD 

sometimes used, but the majority of designs employ some form 
of bolted clamp. The form adopted for the experimental ties 
used by the :N'. Y. C. & H. R. E.R. (see Plate XYII) is especially 
ingenious in the method used to vary the gauge or allow for 
inaccuracies of manufacture. Plate XYII shows some of the 
methods of fastening adopted on the principal types of ties. 

222. Cost. The cost of metal cross-ties in Germany averages 
about 1.6 c. per pound or about $1.60 for a 100-lb. tie. The 
ties manufactured for the IST. Y. C. & H. P. P.P. in 1892 
weighed about 100 lbs. and cost $2.50 per tie, but if they had 
been made in larger quantities and with the j^resent price of 
steel the cost would possibly have been much lower. The 
item of freight from the place of manufacture to the place where 
used is no inconsiderable item of cost with some roads. Metal 
<jross-ties have been used by some street railroads in this country. 



PLATE XVIL 




Metal Ties. 



{Tofacfipnge2A(\ ) 



§ 224. TIES. 241 

Those used on the Terre Haute Street Kailway weigli GO pounds 
and cost about 06 c. for the tie, or 74 c. per tie with tlie 
fastenings. 



223. Bowls or plates. As mentioned before, over 12000 
miles of railway, chiefly in British India and in the Argentine 
Republic, are laid with this form of track. It consists essentially 
■of large cast-iron inverted " bowls" laid at intervals under each 
rail and opposite each other, the opposite bowls being tied 
together with tie-rods. A suitable chair is riveted or bolted on 
to the top of each bowl so as to properly hold the rail. Being 
made of cast iron, they are not so subject to corrosion as steel 
or wrought iron. They have the advantage that when old and 
worn out their scrap value is from 60 to 80^ of their initial 
cost, while the scrap value of a steel or wrought-iron tie is 
practically nothing. Failure generally occurs from breakage, 
the failures from this cause in India being about 0.4 per cent 
per annum. They weigh about 250 lbs. apiece and are there- 
fore quite expensive in first cost and transportation charges. 
There are miles of them in India which have already lasted 
25 years and are still in a serviceable condition. Some illustra- 
tions of this form of tie are show^n in Plate XYII. 



224. Longitudinals."'^ This form, the use of which is con- 
flned almost exclusively to Germany, is being gradually replaced 
on many lines by metal cross- ties. The system generally con- 
sists of a compound rail of several parts, the u])per bearing rail 
Leing very light and supported throughout its length by other 
rails, which are suitably tied together with tie-rods so as to 
maintain the proper gauge, and which have a sufficiently broad 

* Altliou^li the discussion of longitudinals might be considered to belong 
more properly to the subject of Rails, yet the essential idea of all designs 
must necessarily be the support of a rail-head on which the rolling stork may 
run, and therefore this form, unused in this country, will be briefly described 
here. 



242 RAILROAD construction; § 224. 

base to be properly supported in the ballast. One great objection 
to this method of construction is the difficulty of obtaining 
proper drainage especially on grades, the drainage having a 
tendency to follow along the lines of the rails. 
The construction is much more complicated on 
sharp curves and at frogs and switches. An- 
%zzzzzzz2zzi, other fundamentally different form of longi- 
FiG. 110. tudinal is the Haarman compound " self -bear- 

ing " rail, having a base 12'^ wide and a height of 8'', the 
alternate sections breaking joints so as to form a practically 
continuous rail. 

Some of the other forms of longitudinals are illustrated in 
Plate XYII. 

For a very complete discussion of the subject of metal ties, 
see the ' ' Report on the Substitution of Metal for AYood in 
Railroad Ties" by E. E. Russell Tratman, it being Bulletin 
1^0. 4, Forestry Division of the U. S. Dept. of Agriculture. 



CHAPTER IX. 



RAILS. 



225. Early forms. The first rails ever laid were wooden 
stringers wliich were used on very short tram-roads around coal- 
mines. As the necessity for a more durable rail increased, 
owing chiefly to the invention of the locomotive as a motive 
power, there were invented successively the cast-iron "fish- 
belly " rail and various forms of wrought-iron strap rails which 
finally developed into the T rail used in this country and the 
double-lieaded rail, supported by chairs, used so extensively in 
England. The cast-iron rails were cast in lengths of about 3 
feet and were supported in iron chairs which were sometimes 
set upon stone piers. A great deal of the first railroad track 
of this country was laid with longitudinal stringers of wood 
placed upon cross-ties, the inner edge of the stringers being 




CAMDEN & AMBOY. STEPHENSON. "PEAR." 

1832. 1338. 



REYNOLDS— 1767. 

Fig. 111.— Early Forms of Rails. 



protected by wrought-iron straps. The " bridge " rails were 
first rolled in this country in 1844. The ''pear" section was 

243 



244 RAILROAD CONSTRUCTION. § 226. 

an approach to the present form, but was very defective on 
account of the difficulty of designing a good form of joint. The; 
" Stevens " section was designed in 1830 by CoL Robert L. 
Stevens, Chief Engineer of the Camden and Amboy Railroad \ 
although quite defective in its proportions, according to the: 
present knowledge of the requirements, it is essentially the pres- 
ent form. In 1836, Charles Yignoles invented essentially the 
same form in England ; this form is therefore known throughout 
England and Europe as the Yignoles rail. 

226. Present standard forms. The larger part of modern 
railroad track is laid with rails which are either " T " rails or 
the double-headed or " bull-headed " rails which are carried in 
chairs. The double-headed rail was designed with a symmetri- 
cal form with the idea that after one head had been worn out 
by traffic the rail could be reversed, and that its life would be 
practically doubled. Experience has shown that the wear of the 
rail in the chairs is very great ; so much so that when one head 
has been w^orn out by traffic the whole rail is generally useless.. 
If the rail is turned over, the worn places, caused by the chairs, 
make a rough track and the rail appears to be more brittle and 
subject to fracture, possibly due to the crystallization that may 
have occurred during the previous usage and to the reversal of 
stresses in the fibers. Whatever the explanation, experience has 
demonstrated the fact. The " bull-headed " rail has the lower 

head only large enough to properly hold 
the wooden keys with which the rail is. 
secured to the chairs (see Fig. 112) and 
furnish the necessary strength. The use 

FiQ.112.— Bull-headed of these rails requires the use of two cast- 
Rail and Chair. -^.^^ ^j^^.^.^ ^^^ ^^^j^ ^.^^ ^^ .^ ^i^-j^^^j ^j^^^ 

such track is better for heavy and fast traffic, but it is more 
expensive to build and maintain. It is the standard form of 
track in England and some parts of Europe. 

Until a few years ago there was a very great multiplicity 
in the designs of "T" rails as used in this country, nearly 
every prominent railroad having its own special design, which 




^226. 



HAILS. 



245 



perhaps differed from that of some otlier road by only a very 
iiiiiiute and insignilicant detail, but which nevertheless wonld 
require a complete new set of rolls for rolling. This certainly 
must haye had a yery appreciable effect on the cost of rails. In 
l^i93, the American Society of Ciyil Engineers, after a very 
exhaustive investigation of the subject, extending over several 
years, having obtained the opinions of tlie best experts of the 
country, adopted a series of sections wliich have been very ex- 
tensively adopted by the railroads of this country. Instead of 
having the rail sections for yarious weights to be geometrically 
similar figures, certain dimensions are made constant, regardless 
of the weight. It was decided that the metal should be dis- 
tributed through the section in the proportions of head 42^, 

web 21^, and flange 37^. The top of the head should have a 
radius of 12''; the top corner radius of head should be f-^" -^ the 




Fig. 113.— Am. See. C. E. Standard Rail Section. 

lower corner radius of head should be ^\" -, the corners of the 
flanges, ^\" radius; side radius of web, 12''; top and bottom 
radii of web corners, J" ; and angles with the horizontal of the 
under side of the head and the top of the flange, 13°. The 
sides of the head are vertical. 

The height of the rail {!)) and the width of the base {C) are 
always made equal to each other. 



246 



RAILHOAD CONSTRUCTION. 



227. 





Weight per Yard. 


40 


45 


50 


55 


60 


65 


70 


75 


80 


85 


90 


95 


100 


A 
B 
C&D 
E 
F 
G 


1 7// 

5 
g 

HI 


2" 

27 
6« 
Oil 
•^TB 
21 
32 

li-s 




Ol// 

4x^B 

23 

3Z 

Oil 

■*B4 

111 

■1B4 


ii 

II 

U'2 


2M" 
4/e 


2iV' 

11 
41 

2y 


241" 

1 7 
32 

4B 

§1 

211 

HI 


2h" 

35 
64 

5 

7 

8 

2f 


2t^b" 

II 
2| 

Iff 


OS// 

5| 

59 

2|| 
Hi 


2H" 

15 

iB 

963 

Hi 


2|" 

T^B 

5| 

HI 



The chief features of disagreement among railroad men 
relate to the radius of the upper corner of the head and the 
slope of the side of the head. The radius (y^g'O adopted for 
the upper corner (constant for all weights) is a little more 
than is advocated by those in favor of " sharp corners " 
who often use a radius of J". On the other hand it is 
much less than is advocated by those who consider that it 
should be nearly equal to (or even greater than) the larger 
radius universally adopted for the corner of 
the wheel-flange. The discussion turns on 
the relative rapidity of rail wear and the wear 
of the wheel-flanws as affected bv the relation 
of the form of the wheel-tread to that of the 
rail. It is argued that sharp rail corners wear 
the wheel-flanges so as to produce sharp 
flanges, which are liable to cause derailment 
Fig 114 — Rela- ^* switches and also to require that the tires 
TioN OF Rail to of enfi^ine-drivers must be more frequently 

WhEEL-TKEAD. ^ ,*- -, X xi • . J- A 1 

turned down to their true lorm. On the 
other hand it is generally believed that rail wear is much less 
rapid while the area of contact between the rail and wheel-flange 
is small, and that when the rail has worn down, as it invariably 
does, to nearly the same form as the wheel-flange, the rail wears 
away very quickly. 

227. Weight for various kinds of traffic. The heaviest rails 
in regular use weigh 100 lbs. per yard, and even these are only 
used on some of the heaviest traflic sections of such roads as the 
N. Y. Central, the Pennsvlvaiiia, the ]^. Y., N. H. & H., and 




§ 228. HAILS. 247 

a few others. Probably the larger part of the mileage of the 
<i0untrj is laid with 60- to 75-lb. rails — cunsidering the fact that 
^' the larger part of the mileage" consists of comparatively 
lio-ht-tratiic roads and mav exclude all the heavy truuk lines. 
Yery light-traffic roads are sometimes laid with 5G-lb. rails. 
Koads with fairly heavy traffic generally use 75- to 85-lb. rails, 
especially when grades are heavy and there is much and sharp 
curvature. The tendency on all roads is toward an increase in 
the weight, rendered necessary on account of the increase in the 
weight and capacity of rolling stock, and due also to the fact that 
the price of rails has been so reduced that it is both better and 
cheaper to obtain a more solid and durable track by increasing 
the weight of the rail rather than by attempting to support a 
weak rail by an excessive nuinber of ties or by excessive track 
labor in tamping. It should be remembered that in buying rails 
the mere weight is, in one sense, of no importance. The im- 
portant thing to consider is the strength and the stiffness. If 
we assume that all weights of rails have similar cross-sections 
(which is nearly although not exactly true), then, since for beams 
of similar cross-sections the strength varies as the citbe of the 
homologous dimensions and the stiffness as the fourth 2^ower\ 
wdiile the area (and therefore the weight per unit of length) 
only varies as the square^ it follows that the stiffness varies as 
the square of the weight, and the strength as the f power of the 
weight. Since for ordinary variations of weight the price ]>er 
ton is the same, adding (say) 10^ to the weight (and cost) adds 
21^ to the stiffness and over 15 fo to the strength. As another 
illustration, using an 80-lb. rail instead of a 75-11). rail adds only 
i^%% to the cost, but adds about 14^ to the stiffness and neai-ly 
\\% to the strength. This shows wdiy heavier rails are mure 
economical and are being adopted even wdien they are not abso- 
lutely needed on account of heavier rolling stock. The stiffness, 
strength, and consequent durability are increased in a much 
greater ratio than the cost. 

228. Effect of stiffness on traction. A very important but 
generally unconsidered feature of a stiff rail is its effect on trac- 



248 RAILROAD CONSTRUCTION. § 229. 

tive force. An extreme illustration of this principle is seen 
when a vehicle is drawn over a soft sandj road. The constant 
compression of the sand in front of the wheel has virtually the 
same effect on traction as drawing the wheel up a grade whose 
steepness depends on the radius of the wheel and the depth of 
the rut. On the other hand, if a wheel, made of perfectly 
elastic material, is rolled over a surface which, while supported 
with absolute rigidity, is also perfectly elastic, there would be a 
forward component, caused by the expanding of the compressed 
metal just behind the center of contact, which would just bal- 
ance the backward component. If the rail was supported 
throughout its length by an absolutely rigid support, the high 
elasticity of the wheel-tires and rails would reduce this form of 
resistance to an insignificant quantity, but the ballast and even 
the ties are comparatively inelastic. When a weak rail yields, 
the ballast is more or less compressed or displaced, and even 
though the elasticity of the rail brings it back to nearly its 
former place, the work done in compressing an inelastic material 
is wholly lost. The effect of this on the fuel account is certainly 
very considerable and yet is frequently entirely overlooked. It 
is practically impossible to compute the saving in tractive power, 
and therefore in cost of fuel, resulting from a given increase in 
the weight and stiffness of the rail, since the yielding of the rail 
is so dependent on the spacing of the ties, the tamping, etc. But 
it is not difficult to perceive in a general way that such an econ- 
omy is possible and that it should not be neglected in considering 
the value of stiffness in rails. 

229. Length of rails. The standard length of rails with most 
railroads is 30 feet. In recent years many roads have been try- 
ing 45-foot and even 60-foot rails. The argument in favor of 
longer rails is chiefly that of the reduction in track-joints, which 
are costly to construct and to maintain and are a fruitful source 
of accidents. Mr. Morrison of the Lehigh Yalley R.R.^ de- 
clares that, as a result of extensive experience with 45-foot rails 

Report, Roadrnasters Association, 1895. 



§ 230. RAILS. 249 

on that road, he finds that they are much less expensive to 
handle, and that, being so long, they can be laid around sharp 
curves without being curved in a machine, as is necessary with 
the shorter rails. The great objection to longer rails lies in the 
difficnltv in allowing for the expansion, which will require, in 
the coldest weather, an opening at the joint of nearly |" for a 
60-foot rail. The Pennsylvania K.R. and the Norfolk and 
Western R.R. each have a considerable mileage laid with GO-foot 
rails. 

230. Expansion of rails. Steel expands at the rate of .0000065 
of its length per degree Fahrenheit. The extreme range of tem- 
perature to which any rail will be subjected will be about 160°, 
or say from — 20° F. to + 140° F. With the above coefficient 
and a rail length of 60 feet the expansion would be 0.0624 foot, 
or about J inch. But it is doubtful Avhether there would ever 
be such a range of motion even if there were such a range of 
temperature. Mr. A. Torrey, chief engineer of the Mich. 
Cent. R.E., experimented with a section over 500 feet long, 
which, although not a single rail, was made " continuous " by 
rio-id splicing, and he found that there was no appreciable addi- 
tional contraction of the rail at any temperature below + ^0 F. 
The reason is not clear, but i\iQ fact is undeniable. 

The heavy girder rails, used by the street railroads of the 
country, are bonded together with perfectly tight rigid joints 
wliich do not permit expansion. If the rails are laid at a tem- 
perature of 60° F. and the temperature sinks to 0°, the rails 
have a tendency to contract .00039 of their length. If this 
tendency is resisted by the friction of the pavement in which the 
rails are buried, it only results in a tension amounting to .00039 
of the modulus of elasticity, or say 10920 pounds per square 
inch, assuming 28 000 000 as the modulus of elasticity. This 
stress is not dangerous and may be permitted. If the tempera- 
ture rises to 120° F., a tendency to expansion and buckling will 
take place, which will be resisted as before by the pavement, 
and a compression of 10920 pounds per square inch will be in- 
duced, which will likewise be harmless. The range of tempera- 



250 



RAILROAD CONSTRUCTION. 



231 



ture of rails which are buried in pavement is much less than 
when thej are entirely above the ground and will probably 
never reach the above extremes. Eails supported on ties which 
are only held in place by ballast must be allowed to expand and con- 
tract almost freely, as the ballast cannot be depended on to resist 
the distortion induced by any considerable range of temperature, 
especially on curves. 

231. Rules for allowing for temperature. Track regulations 
generally require that the track foremen shall use iron {not 
wooden) shims for placing between the ends of the rails while 
splicing them. The thickness of these shims should vary with 
the temperature. Some roads use such approximate rules as the 
following : " The proper thickness for coldest weather is ^^ of an 
inch ; during spring and fall use i of an inch, and in the very 
hottest weather -^^ of an inch should be allowed." This is on 
the basis of a 30-foot rail. When a more accurate adjustment 
than this is desired, it may be done by assuming some veiy high 
temperature (120° to 150° F.) as a maximum, when the joints 
should be tight; tlien compute in tabular form the spacing for 
each temperature, varying by 20°, allowing 0".01:68 (almost 
exactly -f-^") for each 20° change. Such a tabular form would 
be about as follows (rail length 30 feet) : 



Temperature 


150° 


180° 


110° 


90° 


70° 


50° 


30° 


10° 


- 10° 


- 30° 


Rail opening. , . 





3 " 

6T 


JL" 
3 2 


9 " 
6¥ 


3 " 

T6 


15" 
64 


9 " 
32 


21" 
64 


3" 

8 


2 7" 



One practical difficulty in the way of great refinement in this 
work is the determination of the real temperature of the rail 
when it is laid. A rail lying in the hot sun has a very nnich 
Mgher temperature than the air. The temperature of the rail 
cannot be obtained even by exposing a thermometer directly to 
the sun, although such a result might be the best that is easily 
obtainable. On a cloudy or rainy day the rail has practically 
the same temperature as the air ; therefore on such days there 
need be no such trouble. 



§ 2'62. KAILS. 251 

232. Chemical composition. About OS to 99.5,^ of the com- 
position of steel rails is iron, but the value of the rail, as a rail, 
is almost wholly dependent upon the large number of other 
chemical elements which are, or may be, present in very small 
amounts. The manager of a steel- rail mill once declared that 
their aim was to produce rails having in them — 



Carbon 0.32 to O.±yjyo 

Silicon 0.04 to 0.06^ 

Phosphorus 0.09 to 0.105^ 

Mano-anese 1.00 to 1.50^ 



&' 



The analysis of 32 specimens of rails on the Chic, Mil. & 
St. Paul K.R. showed variations as follows: 

Carbon 0.211 to 0.52^ 

Silicon 0.013 to 0.256^ 

Phosphorus 0.055 to 0.181^ 

Manganese 0.35 to 1.63^ 

These quantities have the same general relative proportions 
as the rail- mill standard given above, the diiferences lying; 
mainly in the broadening of the limits. Increasing the percent- 
age of carbon by even a few hundredths of one per cent makes- 
the rail harder, but likewise more brittle. If a track is well 
ballasted and not subject to heaving by frost, a harder and more; 
brittle rail may be used without excessive danger of breakage,, 
and such a rail will wear much lon^rer than a softer toiiirher 
rail, although the softer tougher rail may be the better rail for 
a road having a less perfect roadbed. 

A small but objectionable percentage of sulphur is some- 
times found in rails, and very delicate analysis will often show 
the presence, in very minute quantities, of several other 
chemical elements. The use of a very small quantity of nickel 
or aluminum has often been suggested as a means of ])roducing 
a more durable rail. The added cost and the uncertaintv of 



252 RAILROAD CONSTRUCTION. § 233. 

the amount of advantage to be gained has hitherto prevented 
the practical nse or manufacture of such rails. 

233. Testing. Cliemical and mechanical testing are both 
necessary for a thorough determination of the value of a rail. 
The chemical testing has for its main object the determination 
of those minute quantities of chemical elements which have such 
a marked influence on the rail for good or bad. The mechanical 
testing consists of the usual tests for elastic limit, ultimate 
strength, and elongation at rupture, detennined from pieces cut 
out of tlie rail, besides a '' drop test." The drop test consists 
in dropping a weight of 2000 lbs. from a height of 16 to 20 
feet on to the center of a rail which is supported on abutm.ents 
placed three or four feet apart. The number of blows required 
to produce rupture or to produce a permanent set of specified 
magnitude gives a measure of the strength and toughness of 
the rail. 

234. Rail wear on tangents. When the wheel loads on a rail 
are abnormally heavy, and particularly when the rail has but 
little carbon and is unusually soft, the concentrated pressure on 

the rail is frequently greater than the elastic 
limit, and the metal "flows " so that the head, 
although greatly abraded, will spread somewhat 
outside of its original lines, as shown in Fig. 
115. The rail wear that occurs on tangents is 
Fig. 115. almost exclusively on top. Statistics show that 

the rate of rail wear on tangents decreases as the rails are more 
W'Orn. Tests of a large number of rails on tangents have shown 
a rail wear averaging nearly one pound per yard per 10 000 000 
tons of trafiic. There is about 33 pounds of metal in one yard 
of the head of an 80 -lb. rail. As an extreme value this may be 
worn down one-half, thus giving a tonnage of 165 000 000 tons 
for the life of the rail. Other estimates bring the tonnage 
down to 125 000 000 tons. Since the locomotive is considered 
to be responsible for one half (and possibly more) of the damage 
done to the rail, it is found that the rate of wear on roads with 
shorter trains is more rapid in proportion to the tonnage, and it 





§ 23o. BAILS. 25-3 

is therefore thought that the life of a rail should be expressed in 
terms of the number of trains. This has been estimated at 
300 000 to 500 000 trains. 

235. Rail wear on curves. On curves the maximum rail wear 
occurs on the inner side of the head of the outer rail, irivinir a 
worn form somewhat as shown in Fig. 116. The dotted line 
shows the nature and progress of the rail wear 
on the inner rail of a curve. Since the pressure 
on the outer rail is somewhat lateral rather than 
vertical, the " flow '' does not take place to the 
same extent, if at all, on the outside, and what- 
ever flow would take place on the inside is Fig. 116. 
immediately worn off by the wheel-flange. Unlike the wear on 
tangents, the wear on curves is at a greater rate as the rail 
becomes more worn. 

The inside rail on curves wears chiefly on top, the same as 
on a tangent, except that the wear is much greater owing to the 
longitudinal slipping of the wheels on the rail, and the lateral 
slipping that must occur when a rigid four-wheeled truck is 
guided around a curve. The outside rail is subjected to a 
greater or less proportion of the longitudinal slipping, likewise 
to the lateral slipping, and, worst of all, to the grinding action 
of the flange of the wheel, which grinds off the side of the 
head. 

The results of some very elaborate tests, made by Mr. 
A. M. Wellington, on the Atlantic and Great Western R.R., on 
the wear of rails, seem to show that the I'ail wear on curves 
may be expressed by tlie formula: " Total wear of rails on a el 
degree curve in pounds per yard per 10 000 000 tons duty 
= 1 -{- O.OScP.''^ "It is not pretended that this fornmla is 
strictly correct even in theory, but several theoretical consider- 
ations indicate that it may be nearly so." According to this 
formula the average rail wear on a 6° curve will be about twice 
the rail wear on a tangent. While this is approximately true, 
the various causes modifying the rate of rail wear (length of 
trains, age and quality of rails, etc.) will result in numerous and 



254 RAILUOAD COIslSTRUCTION. § 236. 

large variations from tlie above formula, which should only be 
taken as indicating an a^^proximate law. 

236. Cost of rails. In 1873 the cost of steel rails was about 
$120 per ton, and the cost of iron rails about $70 per ton. 
Although the steel rails were at once recognized as superior to 
iron rails on account of more uniform wear, they were an 
expensive luxury. The manufacture of steel rails by the Ues- 
semer process created a revolution in prices, and they have 
steadily dropped in price until, during the last few years, steel 
rails have been manufactured and sold for $22 per ton. At 
such prices there is no longer any demand for iron rails, since 
the cost of manufacturing them is substantially the same as that 
of steel rails, while their durability is unquestionably inferior to 
that of steel rails. 



CHAPTEE X. 

RAIL- FASTENINGS. 

RAIL-JOINTS. 

237. Theoretical requirements for a perfect joint. A perfect 
rail- joint is one that has the same strength and stifness — no more 
and ho less — as the rails which it joins, and which will not 
interfere with the regular and uniform spacing of ties. It 
should also be reasonably cheap both in first cost and in cost of 
maintenance. Since the action of heavy loads on an elastic rail 
is to cause a wave of translation in front of each wheel, any 
change in the stiffness or elasticity of the rail structure will 
cause more or less of a shock, which must be taken up and 
resisted by the joint. The greater the change in stiffness the 
greater the shock, and the greater the destructive action of the 
shock. The perfect rail- joint must keep both rail ends trulv in 
line both laterally and vertically, so that the flange or tread of 
the wheel need not jump or change its direction of motion sud- 
denly in passing from one rail to the other. A consideration of 
all the above requirements will show that only a perfect weldino' 
of rail-ends would produce a joint of uniform strength and stiff- 
ness which would give a uniform elastic wave ahead of each 
wheel. As welding is impracticable for ordinary railroad work 
(see § 230), some other contrivance is necessary which will 
approacli this ideal as closely as may be. 

238. Efficiency of the ordinary angle-bar. Throughout the 
middle portion of a rail the rail acts as a continuous girder. If 
we consider for simplicity that the ties are unyielding, the deflec- 
tion of such a continuous girder between the ties will be but 

255 



256 RAILROAD CON'STRUCTIO:sr. § 239. 

one-fourtli of tlie deflection that would be found if the rail were 
cut half-way between the ties and an equal concentrated load 
were divided equally between the two unconnected ends. The 
maximum stress for the continuous girder would be but one-half 
of that in the cantilevers. Joining these ends with rail-joints 
will give the ordinary "suspended" joint. In order to main 
tain uniform strength and stiffness the angle-bars must supply 
the deficiency. These theoretical relations are modified to an 
unknown extent by the unknown and variable yielding of the ties. 
From some experiments made by the Association of Engineers 
of Maintenance of Way of the P. R.K.^ the following deduc- 
tions were made : 

1. The capacity of a "suspended " joint is greater than that 
of a "supported" joint — whether supported on one or three 
ties. (See § 240.) 

2. That (with the particular patterns tested) the angle-bars 
alone can carry only 53 to 56/^ of a concentrated load placed 
on a joint. 

3. That the capacity of the whole joint (angle-bars and rail) 
is only 52.4:^ of the strength of the unbroken rail. 

4. That the ineffectiveness of the angle-bar is due chiefly to 
a deficiency in compressive resistance. 

Although it has been universally recognized that the angle- 
bar is not a perfect form of joint, its simplicity, cheapness, and 
reliability have caused its almost universal adoption. Within a 
very few years other forms (to be described later) have been 
adopted on trial sections and have been more and more extended, 
until their present use is very large. The present time (1900) is 
evidently a transition period, and it is quite probable tliat within a 
very few years the now common angle-plate will be as unknown 
in standard practice as the old-fashioned "fish-plate" is at the 
present time. 

239. Eifect of rail gap at joints. It has been found that the 
jar at a joint is due almost entirely to the deflection of the joint 

* Roadmasters Association of America — Reports for 1897. 



§ 240. RAIL-FASTENINQS. 257 

and scarcely at all to the small gap required for expansion. 
This gap causes a drop equal to the versed sine of the arc hav- 
ing a chord equal to the gap and a radius equal to the radius of 
the wheel. Taking the extreme case (for a 30-foot rail) of a f " 
gap and a 33" freight-car wheel, the drop is about yijVtt"- ^^^ 
order to test how much the jarring at a joint is due to a gap be- 
tween the rails, the experiment was tried of cutting shallow 
notches in the top of an otherwise solid rail and running a loco- 
motive and an inspection car over them. The resulting jarring 
was practically imperceptible and not comparable to the jar pro- 
duced at joints. Xotwithstanding this fact, many plans have 
been tried for avoiding this gap. The most of these plans con- 
sist essentially of some form of compound rail, the sections 
breaking joints. (Of course the design of the compound rail 
has also several other objects in view.) In Fig. 117 are shown a 





Fig. 117.— CoMrouxD Ratl Section^s. 

few of the very many designs which have been proposed. These 
designs have invariably been abandoned after triaL Another 
plan, which has been extensively tried on the Lehigh Yalley 
U.K., is the use of mitered joints. The advantages gained by 
their use are as yet doubtful, while the added expense is unques- 
tionable. The " Eoadmasters Association of America" in 1S95 
adopted a resolution recommending mitered joints for double 
track, l)ut their use does not seem to be growinir. 

240. "Supported," "suspended," and "bridge" joints. In a 
supported joint the ends of the rails are on a tie. If the angle- 
plates are short, the joint is entirely supported on one tie ; if 
very long, it may be possible to place three ties under one angle- 
bar and thus the joint is virtually supported on three ties rather 
than one. In a suspended joint the ends of the rails are midway 
between two ties and the joint is supported by the two. There 



258 RAILROAD CONSTRUCTION. § 241. 

have always been advocates of both methods, but suspended joints 
are more generally used than supported joints. The opponents of 
three-tie joints claim that either the middle tie will be too 
strongly tamped, thus making it a supported joint, or that, if 
the middle tie is weakest, the joint becomes a very long (and 
therefore weak) suspended joint between the outer joint-ties, or 
that possibly one of the outer joint-ties gives way, thus breaking 
the angle-plate at the joint. Another objection which is urged 
is that unless the bars are very long (say tti inches, as used on 
the Mich. Cent. E.K.) the ties are too close for proper tamp- 
ing. The best answer to these objections is the successful use 
of these joints on several heavy-traffic roads. 

" Bridge "-joints are similar to suspended joints in that the 
joint is supported on two ties, but there is the important differ- 
ence that the bridge- joint supports the rail from underneath and 
there is no transverse stress in the rail, whereas the supported 
joint requires the combined transverse strength of both angle- 
bars and rail. A serious objection to bridge- joints lies in the 
fact of their considerable thickness between the rail base and the 
tie. When joints are placed " staggered '* rather than '' oppo- 
site " (as is now the invariable standard practice), the ties sup- 
porting a bridge-joint must either be notched down, thus 
w^eakening the tie and promoting decay at the cut, or else the 
tie must be laid on a slope and the joint and tlie opposite rail 
do not get a fair bearing. 

241. Failures of rail-joints. It has been observed on double- 
track roads that the maximum rail wear occurs a few inches be- 
yond the rail gap at the joint in the direction of the traffic. On 
single-track roads the maximum rail wear is found a few inches 
each side of the joint rather than at the extreme ends of the rail, 
thus showing that the rail end deflects down under the wheel 
until (with fast trains especially) the wheel actually jumps the 
space and strikes the rail a few inches beyond the joint, the 
impact producing excessive wear. This action, which is called the 
"drop," is apt to cause the first tie beyond the joint to become 
depi'essed, and unless this tie is carefully watched and main- 



242. 



RAIL-FA STEISINOb. 



259 



tained at its proper level, the stresses in the aiigle-l)ar may 
actually become rev^ersed and the bar may break at the tu]). The 
angle-bars of a suspended joint are normally in coinpression at 
the top. The mere reversal of the stresses would cause the bars 




Fig. 118. — Eh-ect of " Wheel Drop " (Exaggerated). 

to give way with a less stress than if the stress were always the 
same in kind. A supported joint, and especially a three-tie 
joint (see § 240), is apt to be broken in the same manner. 

242. Standard angle-bars.— An angle-bar must be so made 
as to closely fit the rails. The great multiplicity in the designs 
of rails (referred to in Chapter IX) results in nearly as great 
variety in the detailed dimensions of the angle-bars. The sec- 
tions here illustrated must be considered only as types of the 
variable forms necessary for each different shape of rail. The 
absolutely essential features required for a fit are (1) the angles 




Fig. 119.— Standard Angle-bar— 80-lb. Rail. M. C. R.R. 

of the upper and lower surfaces of the bar where they fit against 
the rail, and (2) the height of the bar. The bolt-holes in the 



260 RAILROAD CONSTRUCTION. § 243. 

bar and rail must also correspond. The holes in the angle-plates 
are elongated or made oval, so that the track-bolts, which are 
made of corresponding shape immediately mider the head, will 
not be turned by jarring or vibration. The holes in the rails 
are made of larger diameter (by about ^") than the bolts, so as 
to allow the rail to expand with temperature. 

243. Later designs of rail-joints. In Plate XYIII are shown 
various designs which are competing for adoption. The most 
j)rominent of these (judging from the discussion in the conven- 
tion of the Roadmasters Association of America in 1897) are 
the " Continuous " and the " Weber." Each of them has been 
very extensively adopted, and where used are universally pre- 
ferred to angle-plates. [N^early all the later designs embody 
more or less directly the principle of the bridge- joint, i.e., sup- 
port the rail from underneath. An experience of several years 
will be required to demonstrate which form of joint best satis- 
fies the somewhat opposed requirements of minimum cost (])oth 
initial and for maintenance) and minimum wear of rails and 
rolling stock. 

TIE-PLATES. 

244. Advantages. (a) As already indicated in § 204, the 
life of a soft-wood tie is very much reduced by "rail-cutting" 
and "spike-killing," such ties frequently requiring renewal 
long before any serious decay has set in. It has been practi- 
cally demonstrated that the "rail-cutting" is not due to the 
mere pressure of the rail on the tie, even with a maximum 
load on the rail, but is due to the impact resulting from 
vibration and to the lono^itudinal workino' of the rail. It has 
been proved that this rail-cutting is practically prevented by 
the use of tie-plates. (h) On curves there is a tendency to 
overturn the outer rail due to the lateral pressure on the side of 
the head. This produces a concentrated pressure of the outer 
edge of the base on the tie which produces rail-cutting and also 
draws the inner spikes. Formerly the only method of guarding 



PLATE XVIII. 




WEIR BOLTED STIFF FROG. 




r7?ff;<«y^yw.'y.^\t 



SECTION THROUGH C-D. SECTION THROUGH A-B. 



ELLIOT PLATE RIVETED FROG. 




SECTION THROUGH PLATE AT POINT. 

Kail Joints and Frogs. 



SECTION THROtIGH SPRING-HOUSING. 



{To face page 260.) 



§ 245. 



RAIL-FASTENiyGS. 



201 



as'ainst this was bv the use of " rail -braces," one pattern of 



which is shown in Fig. 12U. 



But it has been found that tie- 




FiG. 120. 

plates serve the purpose even better, and rail-braces have been 
abandoned where tie-plates are used, {c) Driving spikes through 
holes in the plate enables the spikes on each side of the rail to 
mutually support each other, no matter in which (lateral) direc- 
tion the rail may tend to move, and this probably accounts in 
large measure for the added stability obtained by the use of tie- 
plates, id) The wear in spikes, called ' ' necking, ' ' caused by 
the vertical vibration of the rail against them, is very greatly 
reduced, {e) The cost is very small compared with the value 
of the added life of the tie, the large reduction in the work of 
track maintenance, and the smoother running on the better track 
which is obtained. It has been estimated that by the use of 
tie-plates the life of hard-wood ties is increased from one to 
three years, and the life of soft-wood ties is increased from three 
to six years. From the very nature of the case, the value of 
tie-plates is greater when they are used to protect soft ties. 

245. Elements of the design. The earliest forms of tie-plates 
were llat on the bottom, but it was soon found that they would 
work loose, allow sand and dirt to o^et between the rail and the 
plate and also between the plate and the tie, which would cause 
excessive wear. Such plates are also apt to produce an objec- 
tionable rattle. Another fault of the earlier designs was the use 
of plates so thin that they would buckle. The latest designs 
have flanges or " teetli " formed on the lower surface which 
penetrate the tie about f" to If". Opinion is still divided on 
the question of whether these teeth should run with the grain 



262 BAILBOAD CONSTRUCTION. § 246. 

or across the grain. If the flanges run with the grain, they 
generally extend the whole length of the tie-plate — as in the 
Wolhaupter design. If the grain is to be cut crosswise, several 
teeth about 1" wide will be used — as in the Goldie design. 




WOLHAUPTER 

Fig. 121. — Tie-plates. 

It is a very important feature that the spike-holes shou/d be 
so punched that the spikes will fit closely to the base of tlie rail. 
Otherwise a lateral motion of the rail will be permitted which 
will defeat one of the main objects of the use of the plate. 

Another unsettled detail is the use of "shoulders" on the 
upi^er surface. On the one hand it is claimed that tlie use of 
shoulders relieves the spikes of side pressure from the rail and 
prevents "necking." On the other hand it is claimed that if 
the plain plate is once properly set with new spikes (at least 
Avith spikes not already necked) the spikes will not neck appre- 
ciably, and that, as the shouldered plates cost more, the additional 
expenditure is unnecessary. 

The above designs should be studied with reference to the 
manner in which they fulfill the requirements which have been 
already stated. As in the case of rail-joints, the best forms of 
tie-plates are of comparatively recent design, and experience 
with them is still insufticient to determine beyond all question 
which designs are the best. 

246. Methods of setting. A very important detail in the 
process of setting the tie-plates on the ties is that the flanges or 
teeth should penetrate the tie as far as desired when the plates 
are flrst put in position. It requires considerable force to press 
the teeth into a tie. In a few cases trackmen have depended on 
the easy process of waiting for passing trains to force the teeth 



§247. 



RAIL-FASTENINGS. 



263 



down. Until tlie teeth arc down the spikes cannot l)e driven 
home, and this apparently clieap and easy process resiiks in loose 
spikes and rails. If the trackmen neglect even temporarily to 
tighten these spikes, it will become impossible to make them 
tight ultimately. The })lates are generally pomided into place 
with a 10- to 16-pound sledge-hanmier. A very good method 
was adopted once during the construction of a bridge when a 
pile-driver was at hand. The bridge-ties were placed nnder the 
pile-hammer. The plates, accurately set to gauge, were then 
forced in by a blow from the 8000-lb. hammer falling 2 or 3 
feet. 

SPIKES. 

247. Requirements. The rails must be held to the ties by a 
fastening wdiich will not only give sufHcient resistance, but which 
will retain its capacity for resistance. It must also be cheap 
and easily applied. The ordinary track-spike fulfills the last 
requirements, but has comparatively small resisting power, com- 
pared with screws or bolts. Worse than all, the tendency to^ 
vertical vibration in the rail produces a series of upward pulls on 
the spike that soon loosens it. When motion has once beo-nn 
the capacity for resistance is greatly reduced, and but little more 
vibration is required to pull the spike out 
so much that redriving is necessary. 
Driving the spike to place again in the 
same hole is of small value except as a 
very temporary expedient, as its holding 
power is then very small. Redriving the 
spikes in new holes very soon " spike-kills " 
the tie. Many plans have been devised to 
increase the holding power of spikes, such 
as making them jagged, twisting the spike, 
swelling the spike at about the center of its 
length, etc. But it has been easily demon- 
strated that the fibers of the wood are gen- ^^<^- ^22. 
erally so crushed and torn by driving such spikes that their 
holding power is less than that of the plain spike. 




264 



RAILROAD CONSTRUCTION. 



248. 





The ordinary spike (see Fig. 122) is made with a square 
cross-section which is uniform through the middle of 
its length, the lower If tapering down to a chisel 
edge, the upper part swelling out to the head. The 
Goldie spike (see Fig. 123) aims to improve this form 
by reducing to a minimum the destruction of the 
fibers. To this end, the sides are made smooth^ the 
edges are clean-cut, and the point, instead of being 
chisel-shaped, is ground down to a pyramidal form. 
Such fiber- cutting as occurs is thus accomplished 
without much crushing, and the fibers are thus 
pressed away from the spike and slightly downward. 
Any tendency to draw the spike will therefore cause 
Fig. 123. the fibers to press still harder on the spike and thus 
increase the resistance. 

248. Driving. The holding power of a spike depends largely 
on how it is driven. If the blows are eccentric and irregular 
in direction, the hole will be somewhat 
enlarged and the holding power largely 
decreased. The spikes on each side of 
the rail in any one tie should not be 
directly opposite, but should be staggered. 
Placing them directly opposite will tend ( ^ 
to split the tie, or at least decrease the 
holding power of the spikes. The direc- 
tion of staggering should be reversed in 
the tAvo pairs of spikes in any one tie ^^^- ^24. Spike-driving. 
(see Fig. 124). This will tend to prevent any twisting of the tie 
in the ballast, which would otherwise loosen the rail from the tie. 

249. Screws and bolts. The use of these abroad is very ex- 
tensive, but 'their use in this country has not passed the experi- 
mental stage. The screws are " wood "-screws (see Fig. 125), 
having large square heads, which are screwed down with a 
track- wrench. Holes, having the same diameter as the hase of 
the screw-threads, should first be bored into the tie, at exactly 
the right position and at the proper angle with the vertical. 



§249. 



RAIL-FASTENINGS. 



265 



A liirlit wooden frame is soinetiiiies used to "niido the aiicer at 
the proper angle. Sometimes the large head of the screw bears 
directly against the base of the rail, as with the ordinary 
spike. Other designs employ a plate, made to tit the 
rail on one side, bearing on the tie on the other side, and 
through which the screw passes. These screws cost 
much more than spikes and require more work to put 
in place, but their holding power is much greater 
and the work of track maintenance is very nmcb 
less. Screw-bolts, passing entirely through the tie, 
liavinir the head at the bottom of the tie and the nut on Fig 125. 
the upper side, are also used abroad. These are quite difficult 
to replace, requiring that the ballast be dug out beneath the tie, 
but on the other hand the occasions for replacing such a bolt 
are comparatively rare, as their durability is very great. The 




^ 



^1 


i 




F'"^ 


' 1 

— ^-J 


ill 



Fig. 126. 



use of screws or bolts increases the life of the tie by the avoid- 
ance of " spike-killing." It is capable of demonstration tliat 
the reduced cost of maintenance and the resulting improvement 
in track would much more than repay the added cost of screws 
and bolts, but it seems impossible to induce railroad directors to 
authorize a large and immediate additional expenditure to make 
an annual saving whose value, although unquestionably consider- 
able, cannot be exactly computed. 



266 



RAILROAD CONSTRUCTION. 



250. 



250. "Wooden spikes." Among the regulations for track- 
lajing given in § 208, mention was made of wooden "spikes," 
or plugs, wliicli are used to fill up the holes wlien spikes are 
withdrawn. The value of the policy of filling up these holes is 
unquestionable, since the expense is insignificant compared with 
the loss due to the quick and certain decay of the tie if these 
holes are allowed to fill with water and remain so. But the 
method of making these plugs is variable. On some roads they 
are "hand-made'- by the trackmen out of otherwise useless 
scraps of lumber, the work being done at odd 
moments. This policy, while apparently cheap, is 
not necessarily so, for the hand-made plugs are ir- 
reofular in size and therefore more or less inefticient. 




gang 



IS 



a track 

they may spend 

which could be 

Since the holes 



It is also quite probable that if 
required to make their own plugs, 
time on these very cheap articles 
more profitably employed otherwise, 
made by the spikes are larger at the top than they are 
near the bottom, the plugs should not be of uniform 
cross section but should be slightly wedge-shaped. 
The " Goldie tie-plug" (see Fig. 127) has been de- 
signed to fill these requirements. Being machine- 
made, they are uniform in size ; they are of a shape 
which will best fit the hole ; they can be furnished of any desired 
wood, and at a cost which makes it a wasteful economy to at- 
tempt to cut them by hand. 



Fig. 127. 



TRACK-BOLTS AND NUT-LOCKS. 

251. Essential requirements. The track-bolts must have 
sufticient strength and must be screwed up tight enough to hold 
the angle-plates against the rail with sufficient force to develop 
the full transverse strength of the angle-bars. On the other 
hand the bolts should not be screwed so tight that slipping may 
not take place when the rail expands or contracts with temperature. 
It would be impossible to screw the bolts tight enough to prevent 



§ 252. liAIL-FA8TENINGS. 267 

slipping chiring the contraction due to a considerable fall of 
temperature on a straight track, but when the track is curved, 
or when expansion takes place, it is conceivable that the resist- 
ance of the ties in the ballast to lateral motion may be less than 
the resistance at the joint. A test to determine this resistance 
was made by Mr. A. Torrey, chief engineer of the Mich. Cent. 
R.R., using 80-lb. rails and ordinary angle-bars, the bolts being 
screwed up as usual. It required a force of about 31000 to 
35000 lbs. to start the joint, which would be equivalent to the 
stress induced by a change of temperature of about 22°. Bnt 
if the central angle of any given curve is small, a comparatively 
small lateral component will be sufficient to resist a compression 
of even 35000 lbs. in the rails. Therefore there Avill ordinarily 
be no trouble about having the joints screwed too tight. The 
vibration caused by the passage of a train reduces the resistance 
to slipping. This vibration also facilitates an objectionable 
feature, viz., loosening of the nuts of the track-bolts. The bolt 
is readily prevented from turning by giving it a form wdiich is 
not circular innnediately under the head and making corre- 
sponding holes in the angle-plate. Square holes would answer 
the purpose, except that the square corners in the holes in the 
angle-plates would increase the danger of fracture of the plates. 
Therefore the holes (and also the bolts, under the head) are 
made of an oval form, or perhaps a square form with rounded 
corners, avoiding angles in the outline. 

The nut-locks should be simple and cheap, should have a life 
at least as long as the bolt, should be effective, and should not 
lose their effectiveness with age. IFany of the designs that 
have been tried have been failures in one or more of these 
particulars, as will ])e described in detail below. 

252. Design of track-bolts. In Fig. 128 is shown a common 
design of track-bolt. In its general form this represents 
die bolt used on nearly all roads, being used not only 
with the common angle-plates, but also with many of the im- 
proved designs of rail- joints. The variations are chiefly a 
general increase in size to correspond with the increased 



268 



RAILROAD CONSTRUCTION. 



253. 




Fig. 128 —Track-bolt. 



weight of rails, besides variations in detail dimensions wliicli 
are frequently unimportant. The diameter is usually f '' to 

y ; 1" bolts are sometimes used for 
the heaviest sections of rails. As 
to length, the bolts should not ex- 
tend more than -J" outside of the 
nut when it is screwed up. If it 
extends farther than this, it is liable 
to be broken olf by a possible derail- 
ment at that point. The lengths used 
vary from 3i^', which may be used 
"f^ with 60 lbs. rails, to 5'', which is 
required with 100-lb. rails. The 
length required depends somewliat on 
the type of nut- lock used. 
253. Design of nut-locks. The designs for nut-locks may be 
divided into three classes : {a) those depending entirely on an 
elastic washer which absorbs the vibration which might other- 
wise induce turning; (Ij) those which jam the threads of the 
bolt and nut so that, when screwed up, the frictional resistance 
is too great to be overcome by vibration ; {c) the ' ' positive ' ' 
nut-locks — those which mechanically hold the nut from turning. 
Some of the designs combine these principles to some extent. 
The ' ' vulcanized fiber ' ' nut-lock is an example of the first 
class. It consists essentially of a rubber washer which is pro- 
tected by an iron ring. When first placed this lock is effective, 
but the rubber soon hardens and loses its elasticity and it is then 
ineffective and worthless. Another illustration of class {a) is 
the use of wooden blocks, generally of 1" to '■2" oak, which 
extend the entire length of the angle- bar, a single piece forming 
the washer for the four or six bolts of a joint. This form is 
cheap, but the wood soon shrinks, loses its elasticity, or decays so 
that it soon becomes worthless, and it requires constant adjust- 
ment to keep it in even tolerable condition. The " Yerona" 
nut-lock is another illustration of class {a) which also combines 
some of the positive elements of class {c). It is made of 



§258. 



RAIL- FASTE^^INGS. 



269 



tempered steel and, as shown in Fig. 129, is warped aiid lias 
sharp edges or points. The warped form furnishes the element 
of elastic pressure when the nut is screwed up. The steel 
beino- harder than the iron of the angle-bar or of the nut, it 
bites into them, owing to the great pressure that must exist 




VERONA 





NATIONAL^ 




JONES 

excelsior-- 

Fig. 129.— Types of Nut-locks. 



Tvhen the washer is squeezed nearly flat, and thus prevents any 
hackward movement, although forward movement (or tighten- 
ing the bolt) is not interfered with. The " National " nut-lock 
is a type of the second class {h), in which, like the " Harvey " 
nut-lock, the nut and lock are combined in one piece. With 
six-bolt ande-bars and 30-foot rails, this means a saving of 2112 
pieces on each mile of single track. The " National " nuts are 
open on one side. The hole is drilled and the thread is cut 
sliijhtly smaller than the bolt, so that when the nut is sci'ewed 



270 BAILROAD CONSTRUCTION. § 253. 

up it is forced slightly open and therefore presses on the threads 
of the bolt with such force that vibration cannot jar it loose. 
Unlike the " j^ational " nut, the *' Harvey" nut is solid, but 
the form of the thread is progressively varied so that the thread 
pinches the thread of the bolt and the friction al resistance to 
turning is too great to be affected by vibration. 

The ''Jones" nut-lock, belonging to class (<?), is a type of 
a nut-lock that does not depend on elasticity or jamming of 
screw-threads. It is made of a thin flexible plate, the square 
part of which is so large that it will not turn after being placed 
on the bolt. After the nut is screwed up, the thin plate is bent 
over so that the re-entrant angle of the plate engages the corner 
of the nut and thus mechanically prevents any turning. The 
metal is supposed to be sufficiently tough to endure without 
fracture as many bondings of the plate as will ever be desired. 
JS^ut-locks of class (d) are not in common use. 



CHAPTEK XI. 

SWITCHES AND CROSSINGS. 

SWITCH CONSTRUCTION. 

254. Essential elements of a switch. Flanges of some sort are 
a necessity to prevent car-wheels from running olf from the rails 
on which they may be moving. But the lianges, although a 
necessity, are also a source of complication in that they require 
some special mechanism which will, when desired, guide the 
wlieels out from the controlling inHuence of the main-line rails. 
This must either he done by raising the wheels high enough 
so that the flanges may pass over the rails, or by breaking the 
continuity of the rails in such a way that channels or "flange 
spaces " are formed through the rails. An oi-dinary stub switch 
breaks the continuity of the main-line rails in three places, two 
of them at the switch- block and one at the frog. The Wharton 
switch avoids two of these breaks by so placing inclined planes 
that the wheels, rolling on their flanges, will surmount these 
inclines until they are a little higher than the rails. Then the 
wheels on the side toward which the switch runs are guided 
over and across the main rail on that side. This rise being ac- 
complished in a short distance, it becomes impracticable to 
operate these switches except at slow speeds, as any sudden 
change in the path of the center of gravity of a car causes very 
destructive jars both to the switch and to the rolling stock. The 
other general method makes a break in one main rail (or both) 
at the switch-block. In both methods the wheels are led to one 
side l)y means of the ' ' lead rails, ' ' and Anally one line of wheels 
passes through the main rail on that side by means of a " frog.'' 
There are some designs by which even this break in the main 
rail is avoided, the wheels being led on.'er the main rail by means 

271 



'212 RAILROAD CONSTRUCTION. § 255. 

■of a sliort movahle rail wliicli is on occasion placed across the 
main rail, but such designs have not come into general use. 

255. Frogs. Frogs are provided with two channel- ways or 
^' flange spaces ' ' through which the flanges of the wheels move. 
Each channel cuts out a parallelogram from the tread area. 
Since the wheel-tread is always wider than the rail, the wing- 
rails will support the wheel not only across the space cut out by 



a 



:::i' 



Fig. 130. — Diagrammatic Design of Frog. 

the channel, but also until the tread has passed the point of the 
frog and can obtain a broad area of contact on the tongue of the 
frog. This is the theoretical idea, but it is very imperfectly 
realized. The wing rails are sometimes subjected to excessive 
wear owing to " hollow treads " on the wheels — owing also to 
the frog being so flexible that the point "ducks" when the 
wheel approaches it. On the other hand the sharp point of the 
frog will sometimes cause destructive wear on the tread of the 
wheel. Therefore the tons^ue of the froo; is not carried out to 
the sharp theoretical point, but is purposely somewhat blunted. 
But the break which these channels make in the continuity of 
the tread area becomes extremely obje';tionable at high speeds, 
being mutually destructive to the rolling stock and to the frog. 
The jarring has been materially reduced by the device of 
"spring frogs" — to be described later. Frogs were originally 
made of cast iron — then of cast iron with wearing parts of cast 
steel, which were fitted into suitable notches in the cast iron. 
This form proved extremely heavy and devoid of that elasticity 
of track which is necessary for the safety of rolling stock 
and track at high speeds. The present universal practice is to 
build the frog up of pieces of rails which are cut or bent as re- 
quired. These pieces of rails (at least four) are sometimes 



§ 256. SWrrCRES AND CROSSINGS 273 

assembled by riveting tbein to a iiat plate, but tbis inetliod is 
now but little used, except for very ligbt work. Tbe usual 
practice is now cbietiy divided between " bolted " and '' keyed " 
frogs. In eacli case tbe space between tbe rails, except a sutH- 
cicnt liange-way, is tilled witb a cast-iron tiller and tbe wbole 
assemblage of parts is suitably bolted or clamped togetber, as is 
illustrated in Plate XVIII. Tbe operation of a spring-rail frog 
is evident from tbe ligure. Since a siding is usually o[)erated at 
slow speed, wliile tbe main track may be operated at fast speed, 
a spring- rail frog will be so set tbat tbe tread is continuous for 
tbe main track and broken for tbe sidinii-. Tliis also means tbat 
tbe spi'ing rail will only be moved by trains moving at a (pre- 
sumably) slow speed on to tbe siding. For tbe fast trains on tbe 
main line sucli a frog is substantially a " fixed " frog and lias a 
tread wbicb is practically continuous. 

256. To find the frog number. Tbe frog number (71) equals 
tbe ratio of tbe distance of any point on tbe tongue of tbe frog 
from tbe tbeoretical point of tbe frog divided by tbe widtb of 
tbe tongue at tbat point, i.e. = he -^ ah (Fig. 130). Tbis 
value may be directly measured by applying any convenient 
unit of measure (even a knife, a sbort pencil, etc.) to some 
point of tbe tongue wliere tbe widtb just equals tbe unit of 
measure, and tlien noting bow many times tbe unit of measure 
is contained in tbe distance from tbat place to tbe tbeoretical 
point. But since c, tbe tbeoretical point, is not so readily 
determinable witb exactitude, it being tbe imaginary inter- 
section of tbe gauge lines, it may be more accurate to measure 
de^ ab^ and lis / tben n^ tbe frog number, = hs -4- {ah -j- de). 
If tbe frog angle be called 7^, tben 

n = he H- ah — hs -^ {ah -\- de) = ^ cot ^F 1 

i.e.. cot \F = 2n. 

257. Stub switches. Tbe use of these, although once nearly 
"universal, has been practically abandoned as turnouts from 
7)iain track except for the poorest and cheapest roads. In some 
States, their use on main track is prohibited by law. Tliey 



274 



BAILKOAD CONSTRUCTION. 



257. 



have the sole merit of cheapness with adaptabihty to the cir- 
cumstances of very light traffic operated at slow speed when a 
considerable element of danger may be tolerated for the sake of 
economy. The rails from ^ to ^ (see Fig. ISl'^) are not fastened 




Fig. 131.— Stub Switch. 

to the ties ; they are fastened to each other by tie-rods which 
keep them at the proper gauge ; at and back of B they are 
securely spiked to the ties, and at A they are kept in place by 
the connecting bar ((7) fastened to the switch-stand. One great 
objection to the switch is that, in its usual form, when operated 
as a trailing switch, a derailment is inevitable if the switch is 
misplaced. The very least damage resulting from such a derail- 
ment must include the bending or breaking of the tie-rods of the 
switch-rail. Several devices have been invented to obviate this 
objection, some of wdiich succeed very well mechanically, 
although their added cost precludes any economy in the total 
cost of the switch. Another objection to the switch is the 
looseness of construction which makes the switches objectionable 
at high speeds. The gap of the rails at the head-block is always 
considerable, and is sometimes as much as two inches. A 



* The student sliould at once appreciate tliat in Fig. 131, as well as in 
nearly all the remaining figures in this chapter, it becomes necessary to use 
excessively large frog angles, short radii, and a very wide gauge in order to 
illustrate the desired principles with figures which are sufficiently small for 
the page. In fact, the proportions used in the figures are such that serious 
mechanical difficulties would be encountered if they were used. These 
difficulties are here ignored because they can he neglected in the proportions 
used in practice. 



§ 258. 



SWITCHES AND CROSSINGS. 



275 



driving-wheel with a load of 12000 to 20000 pounds, jumping 
this gap with any considerable velocity, will do innnense damage 
to the farther rail end, besides producing such a stress in the 
construction that a breakage is rendered quite likely, and such a 
breakage might have very serious consequences. 

258. Point switches. The essential principle of a point 
switch is illustrated in Fig. 132. As is shown, one main rail 
and also cue of the switch-rails is unbroken and inmiovable. 




Fig. 132. — Point Switch. 

The other main rail (from A to F) and the corresponding 
portion of the other lead rail are snbstantially the same as in a 
stub switch. A portion of the main rail {AB) and an equal 
length of the opposite lead rail (usually 15 to 2-1 feet long) are 
fastened together by tie-rods. The end at A is jointed as usual 
and the other end is pointed, both sides being trimmed down 
so that the feather edge at B includes the web of the rail. In 
order to retain in it as much strength as pos- 
sible, tlie point-rail is raised so that it rests 
on the base of the stock- rail, one side of the 
l)ase of the point-rail being entirely cut away. 
As may be seen in Fig. 133, although the 
influence of the point of the rail in moving 
the wheel-flange away from the stock-rail is 
really zero at that point, yet the rail has all 
the strength of the web and about one-half 
that of the base — a very fair angle-iron. 
The planing runs back in straight lines, until at about six or 
seven feet back fromx the point the full width of the head is 




Fig. 133. 



276 



RAILROAD CONSTRUCTION. 



259. 



obtained. The full width of the base will only be obtained at 
about 13 feet from the point. An 80-lb. rail is 5 inclies 




Fig. 134.— Ground Lever foe Throwing a Switch. 
wide at the base. Allowing I" more for a spike between 
the rails, this gives ^l" as the minimum width between rail 
centers at the joint. The minimum angle of 
the switch-point (using a 15-foot point rail) 
is therefore tlie angle w^iose tangent is 

5.75 
-.g yr -.9 ~ -03914, which is the tangent of 

1° 50'. Switch-rails are sometimes used with 
a length of 24 feet, which reduces the angle 
of the switch- point to 1° 09'. 



259. Switch-stands. The simplest and 
cheapest form is the " ground lever," wdiich 
has no target. The radius of the circle de- 
scribed by the connecting-rod pin is precisely 
one-half the throw. From the nature of the 
motion the device is practically self-locking in 
either position, padlocks being only used to 
prevent malicious tampering. The numerous 
designs of upright stands are always combined 
with targets, one design of which is illustrated 
in Fig. 135. When the road is ecpiipped 
with interlocking signals, the switch-throw 
meclianisra forms a part of the design. 
^^^" ^'^^- 260. Tie-rods. These are fastened to tlie 

webs of the rails by means of lugs which are bolted on, there 



~(^ 



§ 261. SWITCHES AND CROSSINGS. 217 

being usually a hinge-joint between the rod and the lug. Four 
such tie-rods are generally necessary. The hrst rod is some- 
times made without hinges, which gives additional stiffness to 
the comparatively w^eak rail-points. The old fashioned tie- rod, 
Iiaving jaws litting the base of the rail, was almost universally 
used in the days of stub switches. One great inconvenience 
in their use lies in the fact that they must be slipped on, one by 
one, over the free ends of the switch-rails. Somethnes the 
higs are fastened to the rail-webs bv rivets instead of bolts. 

iR 



^K: 




i 



33 d^ 



Fig, 186. — FouMS of Tie-kods. 

261. Guard-rails. As shown in Figs. 131 and 132, guard- 
rails are used on both the main and switch tracks opposite the 
frog-point. Their function is not only to prevent the possibil- 
ity of the wheel-flanges passing on the wrong side of the frog- 
point, but also to save the side of the frog-tongue from exces- 
sive wear. The necessity for their use may be realized by 
noting the very apparent wear usually found on the side of the 
head of the guard-rail. The flange- way space between the 
heads of the guard-rail and wheel-rail therefore becomes a 
definite quantity and should equal about two inches. Since this 
is less than the space between the heads of ordinary (say 
80-pound) rails when placed base to base, to say nothing of the 
f necessary for spikes, it becomes necessary to cut the flange 
of the guard-rail. The length of the rail is made from 10 to 
15 feet, the ends being bent as shown in Fig. 132, so as to 



278 



RAILROAD CONSTRUCTION. 



§2t)2. 



prevent the possibility of the end of the rail being struck by a. 



wheel -flange. 



MATHEMATICAL DESIGN OF SWITCHES. 

In all of the following demonstrations regarding switches, 
turnouts, and crossovers, the lines are assumed to represent the 
gauge-lines — i.e., the lines of the inside of the head of the 
rails. 

262. Design with circular lead-rails. The sim^^lest method 

is to consider that the lead -rails curve 
out from the main track -rails by arcs 
of circles which are tangent to the main 
rails and which extend to the frog-jDoint 
F. The simple curve from D to F \^ 
of such radius that (^' -|- ^g) vers 7^= g^ 
in which F = the frog angle, g — 
gauge, Z = the "lead" (^F), and 
r = the radius of tne center of the 
Fig. 137. switch rails. 




'^' + ig = 



9 



vers I^ 



Also BF-^ BD = cotiF', BD = g; BF=Z. 



(74) 



Also 



Z 

Z 

QT 



g cot ^F . 

(;• + *</) sin F; 

2r sin \F. 



(75) 
(76) 
(77) 



These formulae involve the angle F. As shown in Table III, 
the angles {F) are always odd quantities, and their trigononietric 
functions are somewhat troublesome to obtain closely with 
ordinary tables. The formulae may be simplified by substitut- 
ing the frog -number n^ from the relation that n =z \ cot \F. 
Since 

\g = L cot F and r -\- ^g = L cosec F^ 



r 



§ 262. SWITCHES AND CROSSINGS. 279 

then r — \L (cot F -\- cosec F) 

= ^g cot h,F io^oi FA;- co^^ec 7^') 

= i^ ^'*-^t^ i-^^5 since (cot a -|- cosec ol) = cot ^^-»r 

= 2^/^^ (78) 

Also L = 2g?i, (TD) 

from which r — ?i X ^- (80) 

These extremely simple relations may obviate altogether the 
necessity for tables, since they involve only the frog-nnmber and 
the gauge. On account of the great simplicity of these rules, 
they are frequently used as they are, regardless of the fact that 
the curve is never a uniform simple curve from switch-block to 
frog. In the first place there is a considerable length of the 
gauge-line within the frog, which is straight unless it is pur- 
posely curved to the proper curve while being manufactured, 
which is seldom if ever done — except for the very large-angled 
frogs used for street-railway work, etc. It is also doubtful 
whether the switch-rails (^xl, Fig. 131) are bent to the com- 
puted curve when the rails are set for the switch. The switch- 
rails of point switches are straight, thus introducing a stretch of 
straio;ht track which is about one-fifth of the total leno-th of the 
lead-rails. The effect of these modifications on the length and 
radius of the lead-rails Avill be developed and discussed in the 
next four sections. 

The throw (t) of a stub switch depends on the weight of the 
rail, or rather on the width of its base. The throw nuist be at 
least f more than that width. The head-block should there- 
fore be placed at such a distance from the heel of the switch {B) 
that the versed sine of the arc equals the throw. These points 
7nvst be opposite on the two rails, but the points on the two rails 
where these relations are exactly true will not be opposite. 
Therefore, instead of considering either of the two radii (r -j- iff} 
and {r — ig), the mean radius r is used. Then (see Fig. 137) 

vers KOQ — t -^ i\ 



280 RAILROAD CONSTRUCTION. 

and the length of the switch- rails is 

QK — r sin KOQ. . 



§263. 



(81) 



These relations develop another disadvantage in the use of a 
stub switch. The required value of BG, using a Xo. 10 frog 
and SO-pound rail, is 30.1 feet — slightly more than a full rail 
leno:th. It would be unsafe to leave so much of the track un- 
spiked from the ties. AYliether this is obviated by spiking down 
a portion of the switch-rails (virtually shortening the lead) or by 
moving the switch-block nearer the heel of the switch (shorten- 
ing the switch- rails), but still maintaining the required throw, 
the theoretical accuracy of the curve is hopelessly lost. 

263. Effect of straight frog-rails. A. portion of the ends of 
the rails of a frog are free and may be bent to conform to the 

switch-rail curve, but there is a con- 
siderable portion which is fitted to the 
cast-iron filler, and this portion is always 
straight. Call the length of this straight 
portion back from the frog-point f 
{= FII, Fig. 138). Then Ve have 

^ + i^ = (^ -/sill F) -^ vers F 




Fig. 138. 



vers 
9 



vers F 



^-/cotii^ 



-2>. 



(82) 



BF= L = {g -/sin F) cot 17^+/ cos F 
= 2gn — y* sin F cot iF-\-f cos F 
= "^^gn -/(I + cos F) +/ cos F 

= ^-gn-f. 

Since r — ig — (Z —f sec F) cot F^ and 
r -\-^g = {L — f cos F) cosec i% 



(83) 



264. 



SWITCHES AND CROSSINGS 



281 



r = iZ (cot jF-\- cosec i^) — if sec F QOt F — \f cos F cosec i'" 

~ ~~ 2 / i^ gill J<^ 



r = Z?i - i/ cot \F 

= Ln — fn. Then from (S3) 

r = 2^71^ — 2f?i 



(8i) 



264. Effect of straight point-rails. The "point switches," 
now so generally used, have straight switcli-rails. This requires 
an angle in the aHgnnient rather than turning off by a tangential 
curve. The angle is, however, very small (between 1° and 2°), 
and the disadvantages of this angle are small compared with the 
very great advantages of the device. 



.^\v- 



?-a 



* 

o 



\ 



MN = /C 



a 




-a 



FM= . 



Fig. 139. 
g - h 



^+ ig = 



smi{F+ a)' 
FiV 



2 sin i(F — a) 
g — k 



2 sin i{F + a) sin i{F 

g - ^- 

cos a — cos F' 



-a) 



(85) 



282 



BAILBOAD CONSTRUCTION. 



§265. 



BF=L=: FM cos i(i^+ a) + DJSr 

= {g- I') cot i{F-\- a) + D^\ 



(86) 



265. Combined effect of straight frog-rails and straight point- 
rails. It becomes necessary in this case to find a curve which 
shall be tangent to both the point-rail and the frog-rail. The 
curve therefore begins at M, its tangent making an angle of (x 
(nsiiallj 1° 50') with the main rail, and runs to H. The central 



1' 



/ FH=/ 

VMDN=a: 
P_^ VHMR=M(F-n:) 



a 




Fig. 140. 

angle of the curve is therefore {F — a). The angle of the chord 
JIM with the main rails is therefore 

i{F^a)+a=.UF+a); 

_ g — f sin F — k 
^^ ~ sinU-^+a) *' 

^-r 29 - ^ sin^(i^- a) 



g — f sin F — h 



2 sin ^X^ + a) sin ^{F — a) 

g — f m\ F — h 
cos a — cos F ^ ' 



ST = 2r sin ^{F - a). 



. (87) 

. (88) 



§266. 



SWITCHES AND CROSSINGS. 



283 



BF = L = lUL cos \{F + ^) +/ cos 7<^ + DN 

=z{g - f sin F - k) cot i{F + a) + / cos 7^^+ D^\ (SO) 

It may be more simple, if (;• -|- ^fj) lias already been com- 
puted, to write 

Z = 2(/' + i(/) sin i{F- a) cos U^+a) +/cos F+ DJV 
= (/' + 4-^)(sin F - sin «') + / cos Z' + DJS', . . (90) 

266. Comparison of the above methods. Computing values 
for r and Z by tlie various methods, on the uniform basis of a 
Is^o. 9 frog, standard gauged' sy\f= 3'. 37, k = 5i"= 0'.479, 
Dy =15' 0", and « = 1° 50', we may tabulate the compara- 
tive results : 





Simple circle 

Curved frog r. 

Curved s\vitch-r. 


§ 263. 
Straitrht frop:-r. 
Curved switch-r. 


§ 204. 

Curved frop:-r. 

Straight switch-r. 


§ 265. 

Straight frog-r. 

Straight switch r. 


r 

Deg. of curve 

L 


762.75 
7" 31' 
84.75 


702.00 

8° 10' 
81.37 


747.48 
7° 40' 
74.00 


681.16 
8° 25' 

72.13 



This shows that the effect of using straight frog-rails and 
straight switch-rails is to sharpen the curve and shorten the lead 
in each case separately, and that the combined effect is still 
greater. The effect of the straight switch -rails is especially 
marked in reducing the length of lead, and therefore Eq. 78 to 
80, although having the advantage of extreme simplicity, can- 
not be used for point-switches without material error. The 
effect of the straio-ht froo^-rail is less, and since it can be mate- 
rially reduced by bending the free end of the frog- rails, the in- 
fluence of this feature is frequently ignored, the frog-rails are 
assumed to be curved and Eq. 85 and 86 are used. (Soe § 276 
for a further discussion of this point.) 



284 



RAILROAD CONSTRUCTION. 



267. 



267. Dimensions for a turnout from the outer side of a curved 
track. In this demonstration the switch-rails will be considered 
as uniformly circular from the switch-points to the frog-point. 




Fig. 141. 
In the triangle FCD (Fig. 141) we have 

{FC+ CD) : (FC- CD) : : tan i{FDC+DFC) : tan i(FDC-DFC) ; 
but i{_FDC+ DFC) = 90° - \d 

and \{FDC - DFC) = iF. 

Also FG+ CD = 2E and FC - CD = g; 

.-. 2B:g: : cot |^: tan iF 
: : cot ^F: tan ^(^ ; 



tan ^6 = ^. 



(91) 



Also OF : FC: : sin 6 : sin ; but cp = {F — 6)\ 

then r -X- -ka = ( M -X- ^a^-. — r-^^- — :^. . . (92) 

(93) 



^7^ = L = 2{R + ^g) sin i^. 



If the curvature of the main track is very sharp or the frog 
angle unusually small, i^may be less than 6-^ in which case the 
center will be on the same side of the main track as C, Eq. 
92 will become (by calling r =. — r and changing the signs) 



(r - \g) = (^ + \g) 



sin B 



sin 



{6-F)- 



(94) 



§267. 



SWITCHES AND CROSSINGS. 



285 



If we call d the degree of curve corresponding to the radius 
r, and D the degree of curve corresponding to the radius i?, also 
d ' the degree of curve of a turnout from a straight track (the 
frog angle F being the same), it may be shown that d — d' — D 
(very nearly). To illustrate we will take three cases, a number 
6 frog (very blunt), a number 9 frog (very commonly used), and 
a number 12 frog (unusually sharp). Suppose 2> = 4° 0'; also 
Z) = 10° 0'; g ^ ■^' Si" = Ir'.TOS. 









D-. 


= 4°. 






Frog: 
number. 












" L " for 
straight track. 


. 


d' - D 


Error. 


L 


6 


12° 


54' 20" 


12' 57' 52" 


0' 03' 32" 


56.57 


56.50 


9 


i 3 


30 27 


3 31 04 


37 


84.85 


84.75 


12 





13 33 


13 36 


03 


112.72 


113.00 








D = 


10° 






Frog: 
number. , 












"i" for 
straight track. 


d 


d'— D 


Error. 


L 


6 


6° 


53' 24" 


6° 57' 52 ' 


0° 04' 28" 


56.66 


56.50 


9 


2 


27 54 


2 28 56 


01 02 


84.86 


84.75 


12 


5 


44 26 


5 40 24 


01 58 


112.91 


113.00 



A brief study of the above tabular form will show that the 
error involved in the use of the approximate rule for ordinary 
curves (-1° or less) and for the usual frogs (about Xo. 9) is really 
insignificant, and that, even for sharper curves (10° or more), 
or for very blunt frogs, the error would never cause damage, 
considering the lower probable speed. In the most unfavorable 
case noted above the change in radius is about Ifc. On account 
of the closeness of the approximation the method is frequently 
used. The remarkable agreement of the computed values of Z 
with the corresponding values for a straight main track (the lead 



286 



RAILROAD CONSTRUCTION. 



268. 



rails circular tlirougliout) shows that the error is insignificant in 
using the more easily computed values. 

268. Dimensions for a turnout from the inner side of a curved 




track. (Lead rails circular throughout.) From Fig. 142 we 
have 

DC+FC\DC -FC::i^ni{DFC+FI)C) : i^n^{DFC- FDC)] 

but k{DFC+ FDC) = 90° - ^6 



and 



^(DFC - FDC) = \F', 
,'. 2i?:^: :cot 1(9: tan ^F 
: : cot ^F: tan |-6'; 



.*. tan 16 



gn 
R' 



(95) 



OF:FC:\du e',^in{F+6). 
ir + J,) = (i? - ^,)g^^^^. 
Z = BF = 2{R - ig) sin id. 



■ (96) 
• (97) 



As in § 267, it may be readily shown that the degree of the 
turnout (d) is nearly the sum of the degree of the main track 
(7>) and the degree (d ') of a turnout from a straight track when 
the frog angle is the same. The discrepancy in this case is 



§ 260. 



SWITCHES AND CROSSINGS. 



287 



somewhat greater tlian in the other, especially when the curva- 
ture of the main track is sharp. If the frog angle is also laro-e, 
the curvature of the turnout is excessively sharp. If the fro<r 
angle is very small, the liability to derailment is great. Turn^ 
outs to the inside of a curved track should therefore be avoided, 
unless the curvature of the main track is small. 

269. Double turnout from a straight track. In Fig. 143 the 
frogs Fi and F, are generally made ecpuil. Then, if^ there are 




Fig. 143. 



uniform curves from B' to F, and from B to i^,, the required 
value of F,n is obtained from 



vers hF^. = -~ — ? , /qo\ 



2-^' m 



r being found from Eq. 78, in which n is the froo- number 
of Fi or F,. ^ 



3IF„, = 7' tan ^F. 



m ) 



but since n„, = ^ cot ^F, 



2-*- »l 5 



MK, = 



r 



2n, 



(99) 



Since vers F, = 



_ 9 



(r + igy 
vers IF„ 



= ^ vers Fi , 



(100) 



288 RAILROAD CONSTRUCTION. § 269. 

Also, since {C,F,)' = {MF^nY + (<^.^^)% we have 




r" + rg + l(f = -^ + 7^\ 
Simplifying and substituting r — ^gn""^ we have 



2/^' + V = ^^'""^ 



"^nm 



a 5 



/t',jj5 



n* 



2n' + i 



1 • 



Dropping the i, which is always insignificant in comparison 
with 27i', we have 

«™ = -2" = '^ X .707 (approx.). . . (101) 

Frogs are usually made with angles corresponding to integral 
values of /i, or sometimes in " half " sizes, e.g. 6, 6|-, 7, 7^, etc. 
If No. 8^ frogs are used for Fi and F^ , the exact frog number 
for i^^ is 6.01. This is so nearly 6 that a ]^o. 6 frog may be 
used without sensible inaccuracy. Numbers 7 and 10 are a 
less perfect combination. If sharp frogs must be used, Sj- and 
12 form a very good combination. 

If it becomes necessary to use other frogs because the right 
combination is unobtainable, it may be done by compounding 
the curve at the middle frog. Fi and F^ should be greater 
than ^F,y^. If equal to J7^„, , the rails would be straight from 
the middle frog to the outer frogs. In Fig. 144, 6^ = Fi~ iF,n^ 
Drawing the chord FiF^, 



KF,F„ = F,-i(>, = F,-iF, + iF„ = i{F, + iF„) ; 



§270. 



SWITCHES A^D CliOSSINQS. 



289 



i^.i^... = 



KF,. 



9 



I-*- m 



in KFiK, 2 sin i{Fi + iF,,) 



sin 



-,;. . . 0^02) 



KFi = KF,,, cot KFiF„, = \(j cot \{F, + W.:) ; (103) 



{r. + 4^) = 



F,F. 



l-*- rn 



2 sin i^ 



^! 

4 sin iCi-', + ii^„,) sin i(JP, - ^F,,) 

^9 



COS ^Fm — COS i^i 



(104) 




Fig. 144. 

If three frogs, all different, miist be used, the largest may be 
selected as F^n ; the radius of the lead rails may be found by an 
inversion of Eq. 98; F,n may be located in the center of the 
tracks by Eq. 99 ; then each of the smaller frogs may be located 
by separate applications of Eq. 102 or 103, the radius being 
determined by Eq. 104. 

270. Two turnouts on the same side. In Fig. 145, let 
0, bisect 0,D. Then {r, + \g) = ^{r, -f- ^g) ; also, 0,0, = 0,F, 
and Fr = F^, 



vers 



F - ^ - ^^ 

-*- ni — . - — : : — 



. (105) 



^F„, = {9\ + i^)sin F„, (106) 



It may readily be shown that the relative values of F^, Fiy 
and F,n are almost identical with those given in § 269 ; as may 



290 



RAILROAD CONSTRUCTION. 



§271. 



be apparent when it is considered that the middle switch may 
be regarded simply as a curved main track, and that, as 




Fig. 145. 

developed in § 267, the dimensions of turnouts are nearly the, 
same whether the main track is straight or slightly curved. 

271. Connecting curve from a straight track. The "con- 
necting curve ' ' is the track lying 
between the frog and the side 
track where it becomes parallel 
to the main track (FS in Fig. 
146 or 147). Call d the distance 
between track centers. The angle 
FO,R = F (see Fig. 146). Call 
t' the radius of the connecting 
curve. Then 

d - g _ 




Fm. 146. 



(^' - ¥j) - 



vers F 



(107) 



FR = (r - ig) sin F. . . (108) 

If it is considered that the distance FI^ consumes too much 
track room, it may be shortened by the method indicated in 
Fig. 151. 

272. Connecting curve from a curved track to the outside. 
When the main track is curved, the required quantities are the 
radius r of the connecting curve from Fto S, Fig. 147, and its 
length or central angle. In the triangle CSF 



OS+CF: CS-OF:: tstn i{OFS-i- CSF) : tmi{CFS -CSF); 



§273. 



SWITCHES AND CROSSINGS. 



291 



but 1{CFS+ CSF) == 90 - ^?/^; and, since the triangle 0,SF 
is isosceles, i{CFS - CSF) = iF; 

o-o 27?+6Z : 6? — ^ :: cot ^^ : tan ^i^ 

: : cot ^F : tan ^ip ; 



1 _ ^^(^ - (/ ) 




Fig. 147. 
From the triangle OO^F we may derive 

r — ig : J2 -\- ig : : sin tp : sin (i^ + ^) ; 

sin i/j 



Also 



ir^3.2(r-i^)sini(i^+^). 



(109) 



(110) 
(111) 



273. Connecting curve from a curved track to the inside. 

As above, it may readily be deduced from the triangle CFS (see 
Fig. 148) that 

{2B - d) : {d - g) :: cot i^p : tan iF, 
and finally that 



2n(d - a) 
tan i.p = ^^j^l 



(112) 



292 RAILROAD CONSTRUCTION. 

Similarly we may derive (as in Eq. 110) 



273. 



Also 



FS = 2{r - kg) sin 4(i^ - ^). 



. (113) 
. (114) 




Fig. 148. 



Two other cases are possible, {a) r may increase until it 

becomes infinite (see Fig. 149), then 
F ziz tp. In such a case we may 
write, by substituting in Eq. 112, 




<2R-d^^n\d-g), 



(115) 



Fig. 149. 



This equation shows the value of 
i?, which renders this case possible 
with the given values of n^ c/, and 
g. (b) ip may be greater than F. 
As before (see Fig. 150) 

2^ — <i : 6? — ^ : : cot -J^ : tan Ji^; 

2n{cl - g) 



§ 274. SWITCHES AND CROSSINGS. 

the same as Eq. 112, but 



rJ^ig^{R -ig) 



sin tp 



sin {ip — F)' 



293 



. (116) 




Fig. 150. 

274. Crossover between two parallel straight tracks. (See 
Fig. 151.) The turnouts are as usual. The crossover track may 
be straight, as shown by the full 
lines, or it may be a reversed 
curve, as shown by the dotted 
lines. The reversed curve short- 
ens the total length of track re- 
quired, but is somewhat objection- 
able. The first method requires 
that both frogs must be equal. 
The second method permits un- 
equal frogs, although equal frogs 
are preferable. The length of 
straiglit crossover track is F^ T. 

F^T sin F,+gcosF, = d-g', 




F,T='^.^^-g cot F,. 
sm J^. ^ 



(117) 



294 RAILROAD CONSTRUCTION. § 274. 

The total distance along the track may be derived as follows : 
DV= 2DF, + F,Y= 2DF, + XT- XF,-, 
XY= {d - g) cot F, ; XF, = g-T-smF,; 

... DV = 2I)F, + {d-g) cot F,-^^^. . . (118) 
If a reversed curve with equal frogs is used, we have 



vers 6 = 



d 



also 



2^, ..... (119) 
DQ = 2/' sin 0. . . . . (120) 




Fig. 152. 
If the frogs are unequal, we will have (see Fig. 152) 

r, vers 6 -\- r^ vers 6 =^ d\ 

vers 6 



d 



+ ) • • • 

also the distance along the track 

B^N = {r, + n) sin 6. . . , 



(121) 



(122) 



275. 



SWITCHES AND CROSSINGS. 



295 



275. Crossover between two parallel curved tracks, (a) Using 
a straight connecting curve. This solution has limitations. If 
one frog (i^,) is chosen, i<\ becomes determined, being a function 
of i^j. If F^ is less than some limit, depending on the width 




Ftg. 153. 

{d) between the parallel tracks, this solution becomes impossible. 
In Fig. 153 assume i^, as known. Then F^H = g sec F,. In 
the triangle TIOF^ we have 

sin HF,0 : sin F,HO :: HO : F,0', 

sin FJIO = cos F, ; HF,0 = 90° + F, ; 

.'„ sin IIF.O = cos F^. 

JIO = I^ + id - ig - (/ sec Fr, F,0 = 7? - ^^ + \g\ 

7^ T^ ^ + ^d — ig — g sec F, ^ ^ ^ 

.-. cos F, - cos F, ^ \ f^ , \ \ . . . (123) 

Pi — hd + ig ^ ^ 



296 



RAILROAD CONSTRUCTION, 



§275. 



Knowing i^,, /9, is determinable from Eq. 91. Fig. 153 shows 
the case where 0^ is greater than F^. Fig. 154 shows the case 
where it is less. The demonstration of Eq. 123 is applicable to 




Fig. 154. 

both figures. The relative position of the frogs F^ and F^ may 
be determined as follows, the solution being applicable to both 
Figs. 153 and 154: 

HOF, = 180° - (90° - F) - (90° + F,) = F, - F,. 
Then 

GF, = 2{R + ^d-\g)^mi{F,-F:). . . . (124) 

Since F^ comes out any angle, its value will not be in general 
that of an even frog number, and it will therefore need to be 
made to order. 

(b) Continuing the switch-rail curves until they meet as a 
reversed curve. In this case F^ and F^ may be chosen at pleasure 
(within limitations), and they will of course be of regular sizes 
and equal or unequal as desired. F^ and F^ being known, 0^ 
and 0^ are computed by Eq. 95 and 91. In the triangle 00^0^ 
(see Fig. 155) 

2{S-00,){S^ 00,) 
^ers rp = oo:zroo^ ' 



in which 



S=i{00, + 00,+ 0,0,); 



§ 275. SWITCHES AjS^D crossings. 

but Or\ = J2+hJ - j\. 



297 



S - Ot\ ^.B + r, - n + id - r, = id; 
^-00, = B + /', -B-id + r, = r^ + >', - id; 




Fig. 155. 



vers ip = 
sin 0(\0, = 



d(r, -\- 7\ — id) 

{R - idTYr'M'^id'^^y ' • • ^^"''^ 

. 00, . B + id-r, 

s,n^,^=sm^^-^-p^; . . (126) 

i^+ 0,0,0; (127) 

2{/2 - id + i(/) sin J(-A - ^, - ^J. . (128) 



298 



RAILROAD CONSTRUCTION. 



276. 



Althougli tlie above method introduces a reversed curve, yet 
it uses up less track than the first method and permits the use of 
ordinary frogs rather than those having some special angle which 
must be made to order. 

276. Practical rules for switch-laying. A consideration of 
the previous sections will show that the formulae are compara- 
tively simple when the lead rails are assumed as circular ; that 
they become complicated, even for turnouts from a straight 
main track, when the effect of straight frog and point rails is 
allowed for, and that they become hopelessly complicated when 
alio wins; for this effect on turnouts from a curved main track. 
It is also shown (§ 267) that the length of the lead is practically 



- — i r 



'" MN=fc 
, FH=/ 

Vhmr^v:; (F-a) 



^ 




Fig. 140. 

the same whether the main track is straight or is curved with 
such curves as are commonly used, and that the degree of curve 
of the lead rails from a curved main track may be found with 
close approximation by mere addition or subtraction. From 
this it may be assumed that, if the length of lead (Z) and the 
radius of the lead rails (r) are computed from Eq. 87 and 90 for 
various fros: ans^les, the same leads mav be used for curved main 
track ; also, that the degree of curve of the lead rails may be 
found by addition or subtraction, as indicated in § 267, and that 
the approximations involved will not be of practical detriment. 



§276. 



SWITCHES AJSD CROSSINGS. 



299 



In accordance with this pkm Table III has been computed from 
Eq. 87, 88, and 90. The leads there given may be used for all 
main tracks straight or curved. The table gives the degree of 
curve of the lead rails for straight main track; for a turnout to 
the inside, add the degree of curve of the main track ; for a 
turnout to the outside, suhtract it. 

If the position of the switch-block is definitely determined, 
then the rails must be cut accordingly ; but when some freedom 
is allowable (wdiich never need exceed 15 feet and may require 
but a few inches), one rail-cutting may be avoided. Mark on 
the rails at B, F, and D ; measure off the length of the switch- 
rails DN\ offset \(j -f li from N for the point S. 
The point H may be located (temporarily) by meas- 
uring along the rail a distance i^7/ (=/) and then 
swinging out a distance of / -^ ii (n being the frog 
number). HT — \(j and is measured at right 
angles to FII. Points for track centers between S 
and T may be laid off by a transit or by the use of a 
string and tape. Substituting in Eq. 31 the value 
of R and of chord (= 8T), w^e may compute x (= 
dh). Locate the middle point d and the quarter 
points a" and c" . Then a" a and c"c each equal 
three-fourths of dl. Theoretically this gives a parabola rather 
than a circle, but the difference for all practical cases is too 
small for measurement. 

Example. Given a main track on a 1° curve ; a turnout to 
the outside, using a number 9 frog; gauge \' 8i" ; /"* = 8'. 87* 
h = hi" : DY = 15' 0" and a = 1° 50'. Then for a straight 
track r would equal 681.16 [d = 8° 25']. For this curved 
track d will be nearly (8° 25' — 4°) = 4" 25', or r will be 
1207.6. Z for the straight track would be 72.20; but since 
the lead is slightly increased (see ^ 2(w) when the turnout is on 
the outside of a curve, Z may hei-e l)e called 72.5. Z7/ = f 
= 3'.37;/-- ii = 3.37--9=0'.375=4".5. 7/, T, and >^ 
may be located as described above. ST may be measured on the 
ground, or it may be computed from Eq. 88, giviuir the value 




«h- 



FiG. 156. 



300 



RAILROAD CONSTRUCTION. 



217. 



of 53.80 feet for straiglit track. Since it is slightly more for a 
turnout to the outside of a curve, it may be called 54.0. Then 

„ (54. oy 

8 X 1297.6 "^ ^'^^^ ^^®*' ^'^^ ^^'' ^^^ ^^" = ^-21 
foot. 



CROSSINGS. 



277. Two straight tracks. When two straight tracks cross 
each other, four frogs are necessary, the angles of two of them 
being supplementary to the angles of the other. Since such 
crossings are sometimes operated at high speeds, they should be 




SECTION ON A-B 



SrCTION^ON OD 

Fm. 157.— Crossing, 



very strongly constructed, and the angles should preferably be 
90° or as near that as possible. The frogs will not in o-eneral 
be "stock" frogs of an even number, especially if the angles 
are large, but must be made to order with the required angles 
as measured. In Fig. 157 are shown the details of such a cross- 
ing. Note the fillers, bolts, and guard-rails. 



§279. 



SWITCHES AND CIIOSSINGS. 



301 



278. One straight and one curved track. Structurally the 
crossino: is about the same as above, but the froir aiiirles are 
all unequal. In Fig. 158, 7? is known, 
and the angle J/, made by the center 
lines of the tracks at their point of inter- 
section, is also known. 

J/ = XCM. SC = n COS M. 
R cos J/+ \(j 



COS i^,= 



7?- 



i^ 



o. .1 1 7^ ^C0SJ/+^^ 

Similarly cos r „ = Trv~y — ■ 

R cos M— \(j 

^ R cos 2f— ^g 

cos F,=z — p—z;, — • 



^(129) 




Fig. 158. 



279. Two curved tracks. The four frogs are unequal, and 
the angle of each must be computed. The radii if, and 7?, are 




:i-- 



Fig. 159. 



known ; also the angle M. r, , 7\ , r, , and 7\ are therefore 
known by adding or subtracting ^g, but the lines are so indi- 



302 



RAILROAD CONSTRUCTION. 



§ 279. 



Trn""" "7"^"'^' ^'^^ '^^ '^^^^ ^^^^» = ^- t^e angle 
MC^C = C[, and the line 0,0, = c. Then 



and 



KC, + 6^) = 90° -J- Jf 



tan iiC - C) = cot iM^^^^' 



C, and C, then become known and 



sm 6, 
In the triangle F, (7. C. , call K« + ^, + r,) = s, ■ then 

vers i^. = ?^^^=i^XfL^Zi) 
Similarly vers i^, = ^(■^^ - n)(.?, - ^j 

vers i^. = ^i^^^ll^k^n) ^ • • (130) 

' l' 3 

vers i^, = ?(fi_ZLZ?)lfi^^ 
In the above equations 



APPENDIX. 



THE ADJUSTMENTS OP INSTRUMENTS. 



The accuracy of instrumental work may be vitiated by any 
one of a large number of inaccuracies in the geometrical rela- 
tions of the parts of the instruments. Some of these relations 
are so apt to be altered by ordinary usage of the instrument that 
the makers have provided adjusting-screws so that the inaccura- 
cies may be readily corrected. There are other possible defects, 
which, however, will seldom be found to exist, provided the 
instrument was properly made and has never been subjected to 
treatment sufficiently rough to distort it. Such defects, when 
found, can only be corrected by a competent instrument maker 
or repairer. 

A WARNING is necessary to those who would test the accuracy 
of instruments, and especially to those whose experience in such 
work is small. Lack of skill in handlintr an instrument will 
often indicate an apparent error of adjustment when tlie real 
error is very different or perhaps non-existent. It is always a 
safe plan wlien testing an adjustment to note the amount of the 
apparent error; then, beginning anew, make another independ- 
ent determination of the amount of tlie error. When two or 
\noYe perfectly independent determinations of such an error are 
made it will generally be found that they differ by an appreciable 
amount. The differences may be due in variable measure to 
careless inaccurate manipulation and to instrumental defects 
which are wholly independent of the particular test being made. 
Such careful determinations of the amounts of the errors are 
generally advisable in view of the next paragraph. 

303 



304 THE ADJUSTMENTS OF INSTRUMENTS. 

Do NOT DISTURB THE ADJUSTING -SCREWS ANY MORE THAN 

NECESSARY. Altliougli metals are apparently rigid, tliey are 
really elastic and yielding. If some parts of a complicated 
mechanism, which is held together largely by friction, are sub- 
jected to greater internal stresses than other parts of the mech- 
anism, the jarring resulting from handling will frequently cause 
a slight readjustment in the parts which will tend to more nearly 
equalize the internal stresses. Such action frequently occurs 
with the adjusting mechanism of instruments. One screw may 
be strained more than others. The friction of parts may pre- 
vent the opposing screw from mimediately taking up an equal 
stress. Perhaps the adjustment appears perfect under these 
conditions. Jarring diminishes the friction between the j)arts, 
and the unequal stresses tend to equalize. A motion takes place 
which, although microscopically minute, is sufficient to indicate 
an error of adjustment. A readjustment, made by unskillful 
hands, may not make the final adjustment any more perfect. 
The frequent shifting of adjusting-screws wears them badly, 
and when the screws are worn it is still more difficult to keep 
them from moving enough to vitiate the adjustments. It is 
therefore preferable in many cases to refrain from disturbing the 
adjusting-screws, especially as the accuracy of the work done is 
not necessarily affected by errors of adjustment, as may be illus- 
trated : 

{a) Certain operations are absolutely unaffected by certain 
-errors of adjustment. 

{J)) Certain operations are so slightly affected by certain small 
errors of adjustment that their effect may properly be neglected. 

(c) Certain errors of adjustment may be readily allowed for 
and neutralized so that no error results from the use of the un- 
adjusted instrument. Illustrations of all these cases will be 
given under their proper heads. 

AD.JUSTMENTS OF THE TRANSIT. 

1. To have the jplate-huhhles m the centei' of the tiibes when 
the axis is vertical. Clamp the upper plate and, with the lower 



THE ADJUSTMENTS OF INSTRUMENTS. 305 

clamp loose, swing the instruineiit so that the plate-bubbles are 
parallel to the lines of opposite leveling-screws. Level up until 
both bubbles are central. Swing the instrument 180°. If the 
bubbles again settle at the center, the adjustment is perfect. If 
either bubble does not settle in the center, move the leveling- 
screws until the bubble is half-icaij back to the center. Then, 
before touching the adjusting-screws, note carefully the position 
of the bubbles and observe whether the bubbles always settle at 
the same place in the tube, no matter to what position the in- 
strument may be rotated. When the instrument is so leveled, 
the axis is truly vertical and the discrepancies between this con- 
stant position of the bubbles and the centers of the tubes measure 
the errors of adjustment. By means of the adjusting-screws 
bring each bubble to the center of the tube. If this is done so 
skillfully that the true level of the instrument is not disturbed, 
the bubbles should settle in the center for all positions of the 
instrument. Under unskillful hands, two or more such trials 
may be necessary. 

When the plates are not horizontal, the measured angle is greater than 
the true horizontal angle by the difference between the measured ancle 
^nd its projection on a horizontal plane. When this angle of inclination 
is small, the difference is insignificant. Therefore when the plate-bubbles 
are very nearly in adjustment, the error of measurement of horizontal 
angles may be far within the lowest unit of measurement used. A smaJl 
€rror of adjustment of the plate-bubble J9e7pm(izm?ar to the telescope will 
affect the horizontal angles by only a small proportion of the error, which 
will be perhaps imperceptible. Vertical angles will be affected by the 
same insignificant amount. A small error of adjustment of the plate- 
bubble iMrallel to the telescope will affect horizontal angles very slightly, 
but will affect vertical angles by the full amount of the error. 

All error due to unadjusted plate-bubbles may be avoided by noting in 
what positions in the tubes the bubbles will remain fixed for all positions 
of azimuth and then keeping the bubbles adjusted to these positions, for 
the axis is then truly vertical. It will often save time to work in this way 
temporarily rather than to stop to make the adjustments. This should 
especially be done when accurate vertical angles are required. 

When the bubbles are truly adjusted, they should remain stationary, 
regardless of whether the telescope is revolved with the upper plate loose 
and the lower plate clamped or whether the whole instrument is revolved, 
the plates being clamped together. If there is any appreciable difference, 



306 THE ADJUSTMENTS OF INSTRUMENTS. 

it shows that the two vertical axes or " centers" of the plates are not con- 
centric. This may be due to cheap and faulty construction or to the exces- 
sive wear that may be sometimes observed in an old instrument originally 
well made. In either case it can only be corrected by a maker. 

2 . To make the revolving axis of the telescoi:>e jyerpendicular 
to the vertical axis of the instrument. This is best tested by 
using a long plumb-line, so placed that the telescope must be 
]3ointed upward at an angle of about 45° to sight at the top of 
the plumb-line and downward about the same amount, if pos- 
sible, to sight at the lower end. The vertical axis of the transit 
must be made truly vertical. Sight at the upper part of the 
line, clamping the horizontal plates. Swing the telescope down 
and see if the cross- wire a<T^ain bisects the cord. If so, the 
adjustment is i^rohaljly perfect (a conceivable exception will be 
noted later) ; if not, raise or lower one end of the axis by means 
of the adjusting-screws, placed at the top of one of the stan- 
dards, until the cross-wire will bisect the cord both at top and 
bottom. The plumb-bob may be steadied, if necessary, by 
hanging it in a pail of water. As many telescopes cannot be 
focused on an object nearer than 6 or 8 feet from the telescope, 
this method requires a long plumb-line swung from a high point, 
which may be inconvenient. 

Another method is to set up the instrument about 10 feet 
from a high wall. After leveling, sight at some convenient 
mark high up on tlie wall. Swing the telescope down and make 
a mark (when working alone some convenient natural mark may 
generally be found) low down on the wall. Plunge the telescope 
and revolve the instrument about its vertical axis and ao^ain sisrht 
at the upper mark. Swing down to the lower mark. If the 
wire again bisects it, the adjustment is perfect. If not, fix a 
point half-way between the two positions of the lower mark. 
The plane of this point, the upper point, and the center of the 
instrument is truly vertical. Adjust the axis to tliese upper and 
lower points as when using the plumb-line. 

3. To inake the line of collimation jperpendicular to the 
revolving axis of the telescope AYitli the instrument level and 



THE ADJUSTMENTS OF INSTRUMENTS. 307 

the telescope nearly horizontal point at some well-defined point 
at a distance of 200 feet or more. Plunge the telescope and 
establish a point in the opposite direction. Turn the whole 
instrument about the vertical axis until it again points at the 
first mark. Again plunge to "direct position" (i.e., with the 
level-tube under the telescope). If the vertical cross- wire again 
points at the second mark, the adjustment is perfect. If not, 
the error is one-fourth of the distance between the two positions 
of the second mark. Loosen the capstan-screw on one side of 
the telescope and tighten it on the other side until the vertical 
wire is set at the one-fourth mark. Turn the whole instrument 
by means of the tangent screw until the vertical wire is midway 
between the two positions of the second mark. Plunge the 
telescope. If the adjusting has been skillfully done, the cross- 
wire should come exactly to the first mark. As an "erecting 
eyepiece " reinverts an image already inverted, the ring carrying 
the cross-wires must be moved in the same direction as the 
apparent error in order to correct that error. 

The necessity for the third adjustment lies principally in the practice 
of producing a line by plunging the telescope, but when this is required to 
be done with great accuracy it is always better to obtain the forward point 
by reversion (as described above for making the test) and take the mean 
of the two forward points. Horizontal and vertical angles are practically 
unaffected by small errors of this adjustment, unless, in the case of 
horizontal angles, the vertical angles to the points observed are very 
different. 

Unnecessary motion of the adjusting-screws may sometimes be avoided 
by carefully establishing the forward point on line by repeated reversions 
of the instrument, and thus determining by repeated trials the exact 
amount of the error. Diffei^ences in the amount of error determined 
would be evidence of inaccuracy in manipulating the instrument, and 
would show that an adjustment based on the first trial would xwohahJy 
prove unsatisfactory. 

The 2d and 3d adjustments are mutually dependent. If either adjust- 
ment is badly out, the other adjustment cannot be made except as 
follows : 

{a) The second adjustment can be made regardless of the third when 
the lines to the high point and the low point make equal angles with the 
horizontal. 



308 TEE ADJUSTMENTS OF INSTRUMENTS. 

(b) The third adjustment can be made regardless of the second when 
the front and rear points are on a level with the instrument. 

When both of these requirements are nearly fulfilled, and especially 
when the error of either adjustment is small, no trouble will be found in 
perfecting either adjustment on account of a small error in the other 
adjustment. 

If the test for the second adjustment is made by means of the plumb- 
line and the vertical cross-wire intersects the line at all points as the tele- 
scope is raised or lowered, it not only demonstrates at once the accuracy 
of that adjustment, but also shows that the third adjustment is either 
perfect or has so small an error that it does not affect the second. 

4. To have the hulible of the telescope-level in the center of 
the tube ivhen the line of colUmation is horizontal. The line of 
collimation should coincide with the optical axis of the telescope. 
If the object-glass and eyepiece have been properly centered, 
the previous adjustment will have brought the vertical cross- 
wire to the center of the field of view. The horizontal cross- 
wire should also be brought to the center of the field of view, 
and the bubble should be adjusted to it. 

a. Peg method. Set up the transit at one end of a nearly 
level stretch of about 300 feet. Clamp the telescope with its 
bubble in the center. Drive a stake vertically under the eye- 
piece of the transit, and another about 300 feet away. Observe 
the height of the center of the eyepiece (the telescope being 
level) above the stake (calUng it a) ; observe the reading of the 
rod when held on the other stake (calling it J) ; take the instru- 
ment to the other stake and set it up so that the eyepiece is ver- 
tically over the stake, observing the height, c ; take a reading on 
the first stake, calling it d. If this adjustment is perfect, then 

a — d =^ h — G^ 

or {a — d) — (b — c) = 0. 

Call (a-d) — Q) — c) = 2m, 

"When m is positive, the line points downward; 
" ??i " negative, " " " upward. 



THE ADJUSTMENTS OF INSTRUMENTS. 309 

To adjust: if the line points up^ sight the horizontal cross- 
wire (by moving the vertical tangent screw) at a point which is 
ra lower, then adjust the bubble so that it is in the center. 

By taking several independent values for «, b, c, and c?, a mean value 
for m is obtained, which is more reliable and which may save much un- 
necessary working of the adjusting-screws. 

h. Using an auxiliary level. When a carefully adjusted 
level is at hand, this adjustment may sometimes be more easily 
made by setting up the transit and level, so that their lines of 
collimation are as nearly as possible at the same height. If a 
point may be found which is half a mile or more away and 
which is on the horizontal cross- wire of the level, the horizontal 
cross- wire of the transit may be pointed directly at it, and the 
bubble adjusted accordingly. Any slight difference in the 
heights of the lines of collimation of the transit and level (say \") 
may almost be disregarded at a distance of ^ mile or more, or, 
if the difference of level w^ould have an appreciable effect, even 
this may be practically eliminated by making an estimated allow- 
ance when sighting at the distant point. Or, if a distant point 
is not available, a level-rod with target may be used at a dis- 
tance of (say) 300 feet, making allowance for the carefully de- 
termined difference of elevation of the two lines of collimation. 

5. Zero of vertical circle. When the line of collimation is 
truly horizontal and the vertical axis is truly vertical, the read- 
ing of the vertical circle should be 0°. If the arc is adjustable, 
it should be brought to 0°. If it is not adjustable, the index 
error should be observed, so that it may be applied to all read- 
ings of vertical angles. 

ADJUSTMENTS OF THE WYE LEVEL. 

1. To make the line of collimation coincide with the center 
of the rings. Point the intersection of the cross-wires at some 
well-defined point which is at a considerable distance. The in- 
strument need not be level, which allows much greater liberty 
in choosing a convenient point. The vertical axis should be 



310 THE ADJUSTMENTS OF INSTRUMENTS. 

clarajDed, and the clips over the wyes should be loosened and raised. 
Rotate the telescope in the wyes. The intersection of the cross- 
wires should be continually on the point. If it is not, it requires 
adjustment. Rotate tlie telescope 180° and adjust one-lialf of 
the error by means of the capstan-headed screws that move the 
cross- wire ring. It should be remembered that, with an erect- 
ing telescope, on account of the inversion of the image, tlie ring 
should be moved in the direction of the ajpjparent error. Adjust 
the other half o:t'the error witli the leveling-screws. Then ro- 
tate the telescope 90° from its usual position, sight accurately at 
the point, and then rotate 180° from that position and adjust 
any error as before. It may require several trials, but it is 
necessary to adjust the ring until the intersection of the cross- 
wires will remain on the point for any position of rotation. 

If such a test is made on a very distant point and again on a point only 
10 or 15 feet from the instrument, the adjustment may be found correct 
for one point and incorrect for the other. This indicates that the object- 
slide is improperly centered. Usually this defect can only be corrected by 
an instrument-maker. If the difference is very small it may be ignored, 
but the adjustment should then be made on a point which is at about the 
mean distance for usual practice — say 150 feet. 

If the whole image appears to shift as the telescope is rotated, it indi- 
cates that the eyepiece is improperly adjusted. This defect is likewise 
usually corrected only by the maker. It does not interfere with instru- 
mental accuracy, but it usually causes the intersection of the cross- wires to 
be eccentric with the held of view. 

2. To make the axis of the level tube parallel to the line of 
collimation. Raise the clips as far as possible. Swing tlie level 
so that it is parallel to a pair of opposite leveling-screws and 
clamp it. Bring the bubble to the middle of the tube by means 
of the leveling-screws. Take the telescope out of the wyes and 
replace it end for end, using extreme care that the wyes are not 
jarred by the action. If the bubble does not come to the center, 
correct one-half of the error by the vertical adjuHting-screws at 
one end of the bubble. Correct the other half by the leveling- 
screws. Test the work by again changing liie telescope end for 
end in the wyes. 



THE ADJUSTMENTS OF INSTRUMENTS. 311 

Care should be taken wliile making this adjustment to see 
that the level-tube is vertically under the telescope. With the 
bubble in the center of the tube, rotate the telescope in the wyes 
for a considerable angle each side of the vertical. If the first 
half of the adjustment has been made and the bubble moves, it 
shows that the axis of the wyes and the axis of the level-tube 
are not in the same vertical plane although both have been made 
horizontal. By moving one end of the level-tube sidewise by 
means of the horizontal screws at one end of the tube, the two 
axes may be brought into the same plane. As this adjustment 
is liable to disturb the other, both should be alternately tested 
until both requirements are complied with. 

By these methods the axis of the bubble is made parallel to 
the axis of the wyes ; and as this has been made parallel to the 
lines of collimation by means of the previous adjustment, the 
axis of the bubble is therefore parallel to the line of collimation. 

3. To make the line of collimation jyerpendicular to tliever- 
tical axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveling carefully 
over one pair of screws, revolve the telescope 180°. If it is not 
level, adjust half of the error by means of the capstan-headed 
screw under one of the wyes, and the other half by the leveling- 
screws. Reverse ao;ain as a test. 

When the first two adjustments have been accurately made, good level- 
ing may always be done by bringing the bubble to the center by means of 
the leveling-screws, at every sight if necessary, even if the third adjust- 
ment is not made. Of course this third adjustment shoukl be made as a 
matter of convenience, so that the line of collimation may be always level 
no matter in what direction it may bo pointed, but it is not necessary to 
stop work to make this adjustment every time it is found to be defective. 

ADJUST^IENTS OF THE DUMPY LEVEL. 

1. To make the axis of the level-tuhe perpendicular to the 
vertical axis. Level up so that the instrument is approximately 
level over both sets of leveling-screws. Then, after leveling 
carefully over one pair of screws, revolve the telescope 180°. If 



312 THE ADJUSTMENTS OF INSTRUMENTS. 

it is not level, adjust one-half of tlie error bj means of the 
adjusting- screws at one end of the bubble, and the other half 
bj means of the leveling-screws. Eeverse again as a test. 

2. To laa'ke the line of collbnation jyerpendicular to the ver- 
tical axis. The method of adjustment is identical with tliat for 
the transit (No. 4, p. 308) except that the cross-wire must be 
adjusted to agree with the level-bubble rather tlian vice versa ^ as 
is the case with the corresponding adjustment of the transit ; 
i.e., with the level-bubble in the center, raise or lower the 
horizontal cross-wire until it points at the mark known to be on 
a level with the center of the instrument. 

If the instrument has been w^ell made and has not been dis- 
torted by rough usage, the cross-wires will intersect at the 
center of the field of view when adjusted as described. If they 
do not, it indicates an error which ordinarily can only be cor- 
rected by an instrument-maker. The error may be due to any 
one of several causes, which are 

{a) faulty centering of object-slide ; 

{b) faulty centering of eyepiece ; 

((?) distortion of instrument so that the geometric axis of 
the telescope is not perpendicular to the vertical axis. If the 
error is only just j)erceptible, it will not probably cause any 
error in the work. 



EXPLANATORY NOTE ON THE USE OF THE TABLES. 



The logarithms here given are "five-place," but the last 
figure sometimes has a special mark over it (e.g., g) which in- 
dicates that one-half a unit in the last place should be added. 
For example : 



the value 
.69586 
.69586 



includes all values between 
.6958575000 + and .6958624999 
.6958635000 + and .6958674999 



The maximum error in any one value therefore does not ex- 
ceed one-quarter of a fifth-place unit. 

AYlien adding or subtracting such logarithms allow a half-unit 
for such a sign. For example: 

.69586 .69586 .69586 

.10841 .10841 .1084! 

.12947 .12947 .12947 



.93374 .93375 .93375 

All other logarithmic operations are performed as usual and 
are supposed to be understood by the student. 

313 



TABLE I.— RADII OF CURVES. 



Deg. 


0° 


10 


2° 


1 


Deg. 


Mill. 


Kadias. 


Log It 


Radius. 


Log R 


Radius. Log R 


Radius. 


LogiJ 1 


Min. 


o 

I 

2 

3 
4 
5 


CO 

343775 

171887 

1 14592 

85944 

68755 


00 
5.53627 
5.23524 
5-05915 
4-93421 
4.83730 


5729.6 
5635-7 

5544-8 
5456.8 
5371.6 
5288.9 


3.75813 
•75095 
.74389 
.73694 
.73010 

.72336 


2864.9 
2841.3 
2818.0 

2795.1 
2772.5 
2750.4 


3- 


45711 
45351 
44993 
44639 
44287 

43939 


1910. I 
1899.5 
1889. I 
1878.8 
1868.6 
1858.5 


3. 


28105 
27864 
27625 

27387 
27151 
26915 




I 
2 

3 

4 
5 


6 

7 
8 

9 

lO 


57296 
491 II 
42972 
38197 
34377 


4.75812 
.69117 

.633I8 
.58203 
.53627 


5208.8 
5131.0 
5055.6 

4982.3 
491 1. 2 


3.71673 
.71026 
.70377 

.69743 
.691I8 


2728.5 
2707.0 
2685.9 
2665.1 
2644 . 6 


3- 


43593 

43249 
42909 

42571 

42235 


1848.5 
1838.6 
1828.8 
1819.I 
1809.6 


3- 


26681 

26448 
26217 

25985 

25757 


6 

7 
8 

9 
10 


II 

12 

13 

14 
15 


31252 
28648 
26444 
24555 

229[8 


4.49488 

.45709 
.42233 
.39014 

.3601 8 


4842.0 
4774.7 
4709.3 
4645.7 
4583-8 


3.68502 

.67895 
.67296 
.66705 
.66122 


2624.4 
2604.5 

2584-9 
2565.6 
2546.6 


3 


41903 
41572 
41245 
40919 
40597 


1800. I 
1790.7 

1781.5 

1772.3 
1763.2 


3 


25529 

25303 
25077 

24853 
24629 


II 

12 

13 
14 
15 


i6 

17 
i8 

19 

20 


21486 

20222 

19099 
18093 

I7I89 


4.33215 
.30582 

.28100 

.25752 
.23524 


4523-4 
4464.7 
4407 . 5 
4351.7 
4297 . 3 


3.65547 
.64979 
.64419 
.63865 

.63319 


2527.9 
2509.5 

2491.3 

2473-4 
2455.7 


3 


.40276 
.39958 
.39642 

.39329 
39017 


1754.2 

1745-3 
1736.5 
1727.8 
1719.1 


3 


24407 
24185 
23967 
23748 
23530 


16 

17 
18 

19 

20 


21 

22 

23 

24 

25 


16370 
15626 
14947 
14324 
13751 


4.2140^ 

.19385 
.17454 
. I 5606 
.13833 


4244.2 

4192.5 
4142.0 

4092.7 
4044.5 


3.62780 
.62247 
.61726 
.61206 
.60685 


2438.3 

2421 . 1 

2404 . 2 

2387.5 
2371.0 


3 


38708 
.38401 

- 38097 
.37794 

- 37494 


1710.6 
1 702 . 1 
1693.7 
1685.4 
1677.2 


3 


23314 
23098 
22884 
.22676 
22458 


21 

22 

23 
24 

25 


26 
27 
28 
29 

30 


13222 
12732 
12278 

II854 
1 1459 


4.12130 
.10491 
.08911 
.07387 
.05915 


3997.5 
3951-5 
3906 . 6 
3862.7 
3819.8 


3.60178 

.59676 
.59186 

.58689 

. 58204 


2354.8 
2338.8 
2323.0 
2307.4 
2292 .0 


3 


37195 
.36899 
. 36604 
.36312 
. 3602 1 


1 669 . I 
1661.0 
1653.0 
1645. I 
1637.3 


3 


22247 
,22037 
.2182^ 
.21619 

21412 


26 
27 
28 
29 
30 


31 

32 

33 
34 
35 


1 1090 

10743 
I04I7 

lOIII 

9822.2 


4.04491 
.03112 

.01776 

4.00479 
3.99221 


3777.9 
3736.8 
3696 . 6 

3657.3 
3618.8 


3.57724 
.57250 
. 56786 
.56316 
.55856 


2276.8 
2261 .9 
2247 . I 
2232.5 
2218. I 


3 


.35733 
.35446 
.35162 

.34879 
.34598 


1629.5 
1621.8 
1614.2 
1606.7 
1599.2 


3 


. 2 1 206 
.21006 
.20795 
.20593 
. 20396 


31 

32 
33 

34 
35 


36 
37 
38 

39 
40 


9549-3 
9291-3 
9046 . 7 
8814.8 
8594.4 


3.97997 
.9680^ 

.95649 
-94521 
.93421 


3581. I 
3544.2 
3508.0 
3472.6 
3437.9 


3.5540T 
.54951 
. 54506 
- 54065 
.53629 


2203.9 
2189.8 
2176.0 
2162.3 
2148.8 


3 


-34318 
.34041 
-33765 
.33491 
.33219 


1591.8 

1584.5 
1577.2 
1570.0 
1562.9 


3 


.20189 

.19988 
.19789 
.19596 
-19392 


36 
37 
38 

39 
40 


41 
42 
43 
44 
45 


8384.8 
8185.2 
7994-8 
7813. I 
7639-5 


3.92349 
.91302 
.90281 
.89282 
-88306 


3403.8 
3370.5 
3337-7 
3305.7 

3274.2 


3-53197 
.52769 

.52345 

.51925 
.51510 


2135-4 
2122.3 
2109.2 
2096.4 
2083.7 


3 


.32949 
.32680 
.32412 

.32147 
.31883 


1555.8 
1548.8 

1541.9 
1535.0 
1528.2 


3 


.19195 
.18999 
.18804 
.18616 
.18417 


41 
42 

43 
44 
45 


46 

47 
48 

49 
50 


7473-4 
7314.4 
7162.0 

7015.9 
6875.6 


3-87352 
.86418 

-85503 
. 84608 

.83731 


3243-3 
3213.0 

3183.2 

3154.0 

3125.4 


3.51098 
.50691 
.50287 
.49886 
.49490 


2071 . I 

2058.7 
2046 . 5 
2034.4 
2022.4 


3 


.31621 
.31360 
.31101 

- 30843 

.30587 


1521.4 

1514.7 
1 508 . 1 

1501.5 
M95-0 


3 


. 18224 
. 18032 
.17842 
.17652 

.17462 


46 
47 
48 
49 
50 


51 
52 
53 
54 
55 


6740.7 
6611 . 1 
6486.4 
6366.3 
6250.5 


3.82871 
.82027 
. 8 1 200 
.80388 
.79591 


3097 . 2 
3069 . 6 
3042.4 
3015.7 
2989-5 


3.49097 
.48707 
.48321 

.47939 
.47559 


2010.6 
1998.9 
1987.3 
1975-9 
1964.6 


3 


.30332 

.30079 

.2982^ 

29577 

29328 


1488.5 
1482.1 

1475-7 
1469.4 
1463.2 


3 


-17274 
.17087 
. 1 6900 
.16714 
.16529 


51 
52 
53 
54 
55 


56 
57 
58 

59 

60 


6138.9 
603 1 . 2 
5927.2 
5826.8 
5729.6 


3.78809 
. 78046 
.77285 
.76542 
.75813 


2963.7 
2938.4 

2913.5 
2889.0 

2864.9 


3.47183 
.46811 
.46441 
.46075 
-45711 


1953.5 
1942.4 

1931.5 
1920.7 

1910. I 


3 


29081 
28835 
28590 

28347 
28105 


1457-0 
1450.8 
1444.7 
1438.7 
1432.7 


3 


-16344 
. 16161 
.15978 
.15796 
15615 


56 
57 
58 

59 
60 



314 



TABLE I.— RADII OF CURVES. 



Deg. 


4° 1 


5" 


1 


1 


Deg. 


Mill. 


liadiiis. 


LOK It 


KiidiuN. Loff It 


ItadiuN. 


Log Jt 


Bud i 118. 


Lo»? Jt 

2.91329 
.91226 
.91123 
.91021 
.909 1 8 
. 908 1 6 


Mill. 


O 

I 
2 

J 

4 
5 


1432.7 
1426.7 
1420.8 
1415.0 
1409.2 

1403 -5 


3-15615 

- 1 5434 
.15255 
.15076 

.14897 
.14720 


1146.3 
II42.5 
1138.7 

1134.9 
II3I .2 

II27.5 


3.05929 
.05784 
.05640 

.05497 
.05354 
.05211 


955-37 
952.72 
950.09 
947.48 
944.88 
942.29 


2.98017 
.97896 

.97776 
.97657 
.97537 
.97418 


819.02 
817.08 
815.14 
813.22 

811 .30 
809 . 40 

807 . 50 
805.61 

803.73 
801.86 
8co . 00 



1 
2 

3 
4 

5 


6 

7 
8 

9 

lO 


1397.8 
1392.1 
1386.5 
1380.9 

1375-4 


3-14543 
.14367 
.14191 
.14017 

.13843 


1123.8 
11 20 . 2 
1 116. 5 
1 112.9 
1109.3 


3.05069 

.04928 

.04787 
. 04646 
.04506 


939.72 
937.16 
934.62 
932.09 
929-57 


2.97300 
.97181 
.97063 
.96945 
.96828 


2 . 907 1 4 
. 906 1 2 
.90511 
. 904 I 
.90309 


6 

7 
8 

9 
10 


1 1 

12 

13 
14 
15 


1369.9 
1364-5 
I359-I 
1353-8 
1348.4 

1343-2 
1338.0 
1332.8 
1327.6 
1322.5 


3 . 1 3669 
.13497 
.13325 
.13154 
.12983 


1105.8 
I 102. 2 
1098.7 
1095.2 
1091.7 


3.04366 
.04227 
.04088 

.03949 
.03811 


927.07 
924.58 
922. 10 
919.64 
917.19 


2 . 967 1 1 
.96594 
.96478 
.96361 
.96246 


798.14 
796.30 
794.46 
792.63 
790.81 


2.90208 
. 90 1 07 
. 9000 / 
.8990? 
. 89807 


1 1 

12 

13 
14 
15 


i6 

17 
i8 

19 

20 


3.12813 
. 1 2644 

.12475 
.12307 
. 1 2 1 40 


1088.3 
1084.8 
1081 .4 
1078.1 
1074.7 


3.03674 
.03537 
.03400 
.03264 
.03128 


914-75 
912.33 
909.92 
907.52 
905-13 


2 . 96 1 36 
.96015 
.95900 
.95785 
.95671 


789.00 
787.20 

785.41 
783.62 
781.84 


2.89708 
. 89608 
.89509 
.89416 
.89312 


16 

17 
18 

19 
20 


21 

22 

23 

24 

25 


1317-5 
1312.4 

1307-4 
1302.5 
1297.6 


3.11974 
.11808 
.11642 
.11477 
.11313 


1071.3 
1068.0 
1064.7 
1061 .4 
1058.2 


3.02992 
.02857 
.02723 
.02589 
.02455 


902 . 76 
900 . 40 
898.05 
895-71 
893-39 


2.95557 
.95443 
.95330 
.95217 
.95104 


780.07 
778.31 

776.55 
774.81 

773.07 


2.89213 
.89115 
.89017 
.88919 
.8882T 


21 

22 

23 
24 

25 


26 
27 
28 
29 

30 


1292.7 

1287.9 
1283. 1 
1278.3 
1273.6 


3.11150 
.10987 
.10825 
. 10663 
. 10502 


1054.9 
1051.7 
1048.5 

1045-3 
1042. 1 


3.02322 
.02189 
.02056 
.01924 
.01792 


891.08 
888.78 
886.49 
884.21 
881.95 


2.94991 
.94879 
.94767 
.94655 
.94544 


771.34 
769.61 
767.90 
766. 19 
764.49 


2.88724 
.88627 
.88536 

.88433 
.88337 


26 
27 
28 
29 
30 


31 

32 

33 
34 
35 


1268.9 
1264.2 
1259.6 
1255.0 
1250.4 


3.10341 
.10182 
.10022 
.09864 
.09703 


1039.0 

1035-9 
1032.8 
1029.7 
1026.6 


3.01661 
.01536 
. 1 400 
.01270 
. 1 1 40 


879.69 

877-45 
875.22 
873.00 
870.80 


2.94433 
.94322 
.94212 
.9410T 
.93991 


762.80 
761. 1 1 

759-43 
757.76 
756. 10 


2.88241 
.88145 
.88049 

.87953 
.87858 


31 
32 
33 
34 

35 


36 

37 

38 

39 
40 


1245.9 
1241.4 
1236.9 
1232.5 
1228.1 


3.09548 
.09391 
.09234 
.09079 
.08923 


1023.5 
1020.5 
1017.5 
IOI4.5 
101 I . 5 


3.01016 
.00882 
.00753 
.00625 
.00497 


868.60 
866.41 
864.24 
862.07 
859.92 


2.93882 
.93772 
.93663 

.93554 
. 93446 


754.44 
752.80 
751.16 

749.52 
747.89 


2.87762 
.87668 
.87573 
.87478 
.87384 


36 
37 
38 
39 
40 


41 
42 
43 
44 
45 


1223.7 
1219.4 
1215.1 
1210.8 
1206.6 


3.08769 
.08614 
.08461 
.08308 

.08155 


1008.6 
1005.6 
1002.7 
999-76 
996.87 


3.00370 

.00242 

3 . 00 1 1 6 

2 . 99989 

.99863 


857-78 
855.65 

853-53 
851.42 

849-32 


2.93337 
.93229 
.93122 

.93014 
.92907 


746.27 
744.66 
743.06 
741.46 
739.86 


2.87290 
.87196 
.87102 
.87008 
. 869 1 5 


41 

42 
43 
44 
45 


46 
47 
48 
49 
50 


1202.4 
1198.2 
1194.0 
1189.9 
1185.8 


3.08003 

.07852 
.07701 

.07550 
.07406 


993-99 
991-13 

988.28 

985-45 
982.64 


2.99738 
.99613 
.99488 

.99363 
.99239 


847-23 

845-15 
843.08 
841 .02 
838.97 


2.92800 
.92693 
.92587 
.92486 
.92374 


738.28 
736.70 

735-13 
733-56 
732.01 


2.86822 
.86729 
.86636 

.86544 
.86451 

2.86359 
.86267 
.86175 
.86084 
.85992 

2.85901 
.85816 
.85719 
.S5629 
.85538 


46 

47 
48 
49 
50 


51 

52 
53 
54 
55 
56 
57 
58 

59 
60 


1181 .7 

II77-7 
1173.6 
1169.7 
1165.7 


3.07251 
.07102 
.06954 
.06806 
.06658 


979 84 
977.06 

974-29 
971-54 
968.81 


2.99115 
.98992 
.98869 

.98746 
.98624 


836.93 
834.90 
832.89 
830.88 
828.88 


2.92269 
.92163 
.92058 

.91953 
.91849 


730.45 
728.91 
727.37 
725.84 
724.31 


51 

52 
53 
54 

55 


1161.8 

1157.9 
1154.0 
II 50. I 
1146.3 


3.0651 T 

.06365 

.06219 

.06074 

1 .05929 


966 . 09 

963-39 
960 . 70 
958.03 
955-37 


2.9850T 
.98380 

.98258 
.98137 
.98017 


826.89 
824.91 
822.93 
820.97 
819.02 


2.91744 

. 9 1 646 

.91536 

.91433 
.91329 


722.79 
721.28 

719.77 
718.27 
716.78 


56 
57 
58 

59 
60 



315 



TABLE I.— RADII OF CURVES. 



Deg. 


is 


° 


9 





1 


1 1 


° 


l»eg. 


Mill. 


Radius. 


Los J« 


Radius. 


Logr a 


Radius. 


Log K 


Kauiu.s. 


L«<i.' 1\ 


3Iin. 


o 


716.78 


2.85538 


637.27 


2 . 80432 


573.69 


2.75867 


521.67 


2.71739 





I 


715.29 


.85448 


636. 10 


.80352 


572.73 


.75795 


520.88 


.71674 


I 


2 


713.81 


.85358 


634.93 


.80272 


571.78 


.75723 


520. 10 


.71608 


2 


3 


712.34 


.85268 


633.76 


.80192 


570.84 


.75651 


519.32 


.71543 


3 


4 


710.87 


.85178 


632.60 


.80113 


569.90 


.75579 


518.54 


.71478 


4 


5 
6 


709.40 


.85089 


631.44 


. 80033 


568.96 


.75508 


517.76 


.71413 


5 


707.95 


2.85000 


630.29 


2.79954 


568.02 


2.75436 


516.99 


2.71348 


6 


7 


706.49 


.84911 


629. 14 


.79874 


567.09 


.75365 


516.21 


.71283 


7 


8 


705.05 


.84822 


627.99 


.79795 


566.16 


.75293 


515.44 


.71218 


8 


9 


703.61 


.84733 


626.85 


.79716 


565-23 


7 ^222 


514.68 


•7II53 


9 


lO 


702. 17 


.84644 


625.71 


.79637 


564.31 


.75151 


513.91 


.71088 
2.71024 


10 


II 


700.75 


2.84556 


624.58 


2.79558 


563.38 


2.75086 


513.15 


II 


12 


699.33 


.84468 


623.45 


.79480 


562.47 


.75009 


512.38 


.70959 


12 


1.3 


697.91 


.84380 


622.32 


.79401 


561.55 


•74939 


511.63 


.70895 


13 


14 


696.50 


.84292 


621 .20 


.79323 


560.64 


.74868 


510.87 


.70831 


14 


15 
i6 


695.09 


. 84204 


620.09 


.79245 


559.73 


.74798 


510. II 


.70767 


15 


693.70 


2.84117 


618.97 


2.79167 


558.82 


2.74727 


509.36 


2.70702 


16 


17 


692.30 


. 84029 


617.87 


.79089 


557.92 


.74657 


508.61 


.70638 


17 


i8 


690.91 


.83942 


616.76 


.79011 


557.02 


.74587 


507.86 


•70575 


18 


19 


689.53 


.83855 


615.66 


.78934 


556.12 


•74517 


507. 12 


.70511 


19 


20 


688.16 


.83768 


614.56 


.78856 


555-23 


. 74447 


C06.38 


.7044/ 


20 


21 


686.78 


2.83682 


613.47 


2.78779 


554.34 


2.74377 


305.64 


2.70383 


21 


22 


685.42 


•83595 


612.38 


.78702 


553-45 


.74307 


504.90 


.70320 


22 


23 


684.06 


.83509 


611.30 


.78625 


552.56 


.74238 


504. 16 


.70257 


23 


24 


682.70 


.83423 


610.21 


.78548 


551.68 


.74168 


503.42 


.70193 


24 


25 

26 


681.35 


.83337 


609 . 1 4 


.7847T 


550.80 


.74099 


502 . 69 


.70130 


25 


680.01 


2.83251 


608 . 06 


2.78395 


549.92 


2.74030 


501 .96 


2.70067 


26 


27 


678.67 


.83166 


606 . 99 


.78318 


549.05 


.73961 


501 .23 


. 70004 


27 


28 


677.34 


.83086 


605.93 


.78242 


548.17 


.73892 


500.51 


.69941 


28 


29 


676.01 


.82995 


604.86 


.78165 


547.30 


.73823 


499.78 


.69878 


29 


30 


674.69 


.82910 


603 . 80 


.78089 


546.44 


•73754 


499 . 06 


.69815 


30 


31 


^n-zi 


2.8282^ 


602.75 


2.78013 


545.57 


2.73685 


498 . 34 


2.69752 


31 


32 


672.06 


.82746 


601 .70 


•77938 


544.71 


.73617 


497.62 


. 69690 


32 


33 


670.75 


.82656 


600.65 


.77862 


543.86 


.73548 


496.91 


.69627 


33 


34 


669.45 


.8257T 


599.61 


.77786 


543.00 


.73480 


496.19 


.69565 


34 


35 


668.15 


.82487 


598.57 


.77711 


542.15 


.73412 


495.48 


.69503 


35 


36 


666.86 


2.82403 


597-53 


2.77636 


541.30 


2.73343 


494.77 


2 . 69446 


36 


37 


665.57 


.82319 


596.50 


.77561 


540.45 


.73275 


494.07 


•69378 


37 


3« 


664.29 


.82235 


595-47 


.77486 


539.61 


.7320^ 


493.36 


.69316 


38 


39 


663.01 


.82152 


594.44 


.77411 


538.76 


.73140 


492.66 


.69254 


39 


40 


661 .74 


.82068 


593-42 


.77336 


537.92 


.73072 


491.96 


.69192 


40 


41 


660.47 


2.81985 


592.40 


2.7726T 


537.09 


2.73004 


491 .26 


2.69131 


41 


42 


659.21 


.81902 


591.38 


.77187 


536.25 


.72937 


490.56 


. 69069 


42 


43 


657.95 


.81819 


590.37 


.77112 


535.42 


.72869 


489.86 


. 69007 


43 


4+ 


656.69 


.81736 


589.36 


.77038 


534.59 


.72802 


489.17 


.68946 


44 


45 


655.45 


.81653 


588.36 


.76964 


533.77 


.72735 


488.48 


.68884 


45 


46 


654.20 


2.81571 


587.36 


2.76890 


532.94 


2.72668 


487.79 


2.68823 


46 


47 


652.96 


.81489 


586.36 


.76816 


532.12 


.72601 


487.10 


.68762 


47 


48 


651.73 


.81406 


585-36 


.76742 


531.30 


.72534 


486.42 


.68701 


48 


49 


650. 50 


.81324 


584-37 


.76669 


530.49 


.72467 


485.73 


.68640 


49 


50 


649.27 


.81243 


583-38 


.76595 


529.67 


.72401 


485.05 


.68579 
2.68518 


50 


51 


648.05 


2.8116] 


582.40 


2.76522 


528.86 


2.72334 


484-37 


51 


52 


646.84 


.81079 


581.42 


. 76449 


528.05 


.72267 


483.69 


.68457 


52 


53 


645.63 


. 80998 


580.44 


.76376 


527.25 


.72201 


483.02 


.68396 


53 


54 


644.42 


.80917 


579-47 


.76303 


526.44 


.72135 


482.34 


.68335 


54 


55 


643.22 


.80836 


578.49 


.76230 


525.64 


.72069 


481.67 


.68275 


55 


56 


642.02 


2.80755 


577.53 


2.76157 


524.84 


2.72003 


48 1 . 00 


2.68214 


56 


57 


640.83 


.80674 


576.56 


.76084 


524.05 


.71937 


480.33 


.68154 


57 


5« 


639.64 


.80593 


575-60 


.76012 


523.25 


.71871 


479.67 


.68094 


58 


59 


638.45 


.80513 


574-64 


.75939 


522.46 


.71805 


479.00 


.68033 


59 


60 


637.27 


.80432 


573-69 


.75867 


521.67 


.71739 


478.34 


.67973 


60 1 



316 



TABLE I.— RADII OF CURVES. 



Kndius. i Log li 



4 
6 
8 

lO 
12 

i6 
_i8_ 

20 

22 

24 
26 
28 

30 
32 

34 
36 



40 
42 
44 
46 

_48_ 

50 
52 

54 
56 
58 



4 , 

6 ' 
8 

10 ' 

12 

14 
16 
18 

20 
22 

24 

26 
28 

30 
32 
34 

36 
-.8 



478.34 2 
477.02 

47571 ' 
474-40 

473- ^o _ 
471.81 2 

470.53 
469.25 

467 • 98 
466.72 



.67973 
.67853 
.67734 
.67614 
■67495 



.67376 

.67258 
.67146 
.67022 
.6690:; 



465.46 
464.21 
462 . 97 

461.73 
!;o 



460 



.66788 
.66671 
.66555 
. 66439 
•66323 



l»eff. 1 Radius. Lo«? It 



u 



10 

12 

14 
16 

20 
22 

24 
26 



459.28 2 
458.06 I 
456.85' 
45565 

454-45 ' 
453^26 2 



.66207 
.66092 

.65977 
.65863 

.65748 



30 
32 
34 
36 
38 



452.07 
450.89 

449-72 
448 . 56 

447 - 40 
446.24 
445.09 

443^95 
442.81 _ 

441 .68 \2. 
440.56 
439-44 
438.331 

437-22 ;_ 

436.12;2 

435-02( 

433^93 I 
432 •84! 
431^76 

430.69 2 
429.62 
428.561 

427.50 

426.44 



.65634 
.65521 
.65407 
.65294 
.6518T 



,65069 

.64957 
,64845 

.64733 
.64622 



,64511 
. 64400 
.64290 
.64180 
. 64070 



40 

42 
44 
46 
48 



410.28 2 
409.31 I 
408 . 34 

407 - 38 j 
406.42 



.61307 
.61205 
.61 102 
. 6 I 000 
. 60898 



405-47 

404-53 
403.58 
402.65 
401.71 



Deg. 



16^ 



400.78 2 
399.86 

398.941 
398.02 

397 



1 1 



; . 60796 
. 60694 
•60593 
.60492 
•60391 

.60291 
. 60 1 96 
. 60096 

- 59990 
.59891 



5 
10 

15 
20 

25 



Kadi us. Log It 



359.26 2.55541 
35742 .55317 



50 
52 

54 

56 
58 



15' 



396. 20 
395-30 
394^40 
393 • 50 
392.61 

391 .72 I2 

390.84 

389.96 

389.08 

388.21 

387^34 2 
386.48 ' 
385.62 

384 • 77 
383^91 



63966 
63851 
63742 
63633 
,63524 



.63416 
.63308 
.63201 

■63093 
.62986 



10 

12 

14 
16 
18 



20 

22 

24 
26 
28 



40 
42 
44 
46 
48 



5-- 

54 
56 
58 



425.40 

424^35 
423-32 
422.28 
421 .26 



420.23 
419.22 
418.20 
417.19 
416. 19 



.62879 
.62773 
.62665 
.62566 
.62454 



32 

34 
36 
38 



415.19 
414.20 
413.21 
412.23 
411.25 



.62349 
.62243 
.62138 
.62034 
.61929 



14^ 1 410.28 2 



61825 
,61721 
,61617 

.61514 
.61416 

61307 



40 
42 
44 
46 
48 



50 

52 
54 
56 
58 



30 

35 
40 

4 5 
50 

55 



355-59 
353-77 1 
351-98 
35021 

348.45 
346.71 
344-99 
343-29 
341 .60 

339-93 



Deg. ; HailiuH. Log Jt 



21 



17 



-55094 
.54872 
- 54652 
• 54432 
2.54214 

-53997 
-53786 

•53565 
-5335' 
' -53138 
338.27 2.52927 
336.64 .52716 
52506 

52297 
52090 
51883 



10 
20 
30 
40 
50 



2.51677 
.51472 
.51269 
. 5 1 066 
. 50864 
• 50663 



319.62 2. 50464 
318.16 .50265 
. 50067 
.49869 
•49673 
•49478 



383.06 2 
382.22 1 
381.38 
380.54 
379.71 



378.88 2 

378^05 

377-23 

376-41 

375.60 



374^79 
373-98 
373-17 
372.37 
371-57 



370.78 

369.99 

369.20 

368.42 

3 67^64 i_ 

366.86 2 

366.09 I 

365^31 

364^55 

363^78 




22° 

10 

20 
30 
40 

i£ 
2.-{° 

10 
20 
30 
40 
50 



274^37 
272.23 
270. 13 
268.06 
26^6 . 02 
264 . 02 



262 . 04 
260 . I o 
258.18 
256.29 

254-43 
252.60 



.49284 
. 49096 
.48898 
.48706 

515 
•48325 

302.94 2.48136 
.47948 
.47766 

-47573 
-47388 
-47203 



250 


79 


249 


01 


247 


26 


245 


53 


243 


82 


242 


14 



24° 

10 

20 
30 
40 

25° 

30 
2«° 

30 



2? 

30 

28° 
30 

2D° 

30 

:}0° 

^o 



81 = 
32 

U 
i5 



56911 
,56819 
.56726 
. 56634 
,56542 



56450 

56358 
56266 

,56175 
, 56084 



l(i 



363-02 
362. 26 
361.51 
360.76 
360.01 



55993 
55902 
55812 
55721 
5563' 



359-26 2.5554T 



2.47018 

-46835 
.46652 

- 4647 1 
.46289 
. 46 1 09 



5 
10 

15 
20 

25 



30 
35 
40 
45 
50 
55 



21 



287.94 
286.76 

285.58 
284.42 
283.27 
282. 12 



280.99 
279.86 
278.75 
277.64 
276.54 
275-45 



274-37 



2-45930 
•45751 
•45573 
-45396 
.45219 

-45044 



My 
37 

38 
39 
40 



41 
42 
43 
44 
45 



2 . 44869 
.44694 
.44521 
•44348 
.4417^' 
- 44004 



2.438 



jj 



4(J 
47 

48 
40 
.")(> 

52 
54 

5(i 

58 

(io 



240 

238 


49 
85 


237 
235 
234 


24 

65 
08 


O T 2 


S4 



2-43833 

■43494 
•43157 
•42823 
.42492 
.42163 

2.41837 

•41513 
.41192 

•40873 
•40557 
•40243 

2 ■39931 
.39622 

• 393 • 5 
.39016 

•38707 

• 3840 7 
2.38109 

•37813 
.37519 
.37227 

.36937 

• 36649 



231 .01 

226.55 

222 . 27 
218. 15 


2.36363 

•35517 
. 34688 

-33875 


214. 18 
210.36 
206.68 
203.13 


2.33078 
•32296 
-31529 
■307/6 



99.70 

96-38 

93-19 

90.09 



- 30037 

- 29316 

-28597 
• 27896 



87. 10 
81 .40 
76.05 
71 .02 

66.28 



61 .80 

57-58 
53-58 

49-79 
46.19 



42.77 
39-52 
36-43 
33-47 
^o . 66 



27 . 97 

25-39 
22.93 

20.57 
18.31 



14.06 
10.13 
06. 50 

03^13 
00.00 



27207 

25863 

24563 

23303 
22083 



20899 

19749 
18633 

17547 
16492 



2 . 1 5464 
-14464 
.13489 
.12539 
•11613 



2. 10709 
.09827 
.08965 
.08124 




317 



TABLE II.— 


TANGENTS. EXTERNAL DISTANCES, AND LONG CHORDS 


FOR M 




1° CURVE. 




1 


A 


Taiigeut 


Ext.Dist. LoiigCh'd 


A 


Taii:;eiit 


Ext.Dist. 


LoiigCh d 


A 


Tansreut 


Ext.Dist. 


LoiigCh'd 


V 


T. 


i;. 


xc. 


T. 


-K. 


LC. 


T. 


E. 


LC. 


50.00 


0.218 


I 00 . 00 


11° 


551.70 


26 . 500 


1098.3 


21° 


1061 .9 


97.58 


2088.3 


lo' 


58.34 


0.297 


116.67 


10 


560. 11 


27.313 


III4.9 


10 


1070.6 


99-15 


2104.7 


20 


66.67 


0.388 


133-33 


20 


568.53 


28. 137 


II3I-5 


20 


1079.2 


100.75 


2121 . 1 


30 


75.01 


0.491 


1 50 . 00 


30 


576.95 


28.974 


I 1 48 . 1 


30 


1087.8 


102.35 


2137.4 


40 


83.34 


0.606 


166.66 


40 


585.36 


29.824 


I164.7 


40 


1 096 . 4 


103.97 


2153.8 


50 


91.68 


0.733 


183.33 


50 


593-79 


30.686 


I181.2 


50 


1105. I 


105.60 


2170.2 


2° 


100.01 


0.873 


199.99 


12= 


602.21 


31.561 


I 197.8 


22° 


1113.7 


107.24 


2186.5 


10 


108.35 


1 .024 


216.66 


10 


610.64 


32.447 


1214.4 


10 


1122.4 


1 08 . 90 


2202.9 


20 


116.68 


1.188 


233-32 


20 


619.07 


33-347 


. 1231 .0 


20 


II3I.O 


110.57 


2219.2 


30 


125.02 


1.364 


249.98 


30 


627.50 


34-259 


1247.5 


30 


II39-7 


112.25 


2235.6 


40 


133.36 


1.552 


266.65 


40 


635-93 


35-183 


1264. I 


40 


1148.4 


113-95 


2251 .9 


50 


141.70 


1.752 


283.31 


50 


644-37 


36.120 


1280.7 


50 


1157.0 


115.66 


2268.3 


3° 


I 50 . 04 


1.964 


299.97 


13° 


652.81 


37.069 


1297.2 


23° 


1165.7 


117.38 


2284.6 


10 


158.38 


2.188 


316.63 


10 


661 .25 


38.031 


I313-8 


10 


1174.4 


119. 12 


2301 .0 


20 


166.72 


2.425! 333-29i 


20 


669.70 


39 . 006 


1330.3 


20 


1183.1 


120.87 


2317.3 


30 


175.06 


2.674 


349-95 


30 


678.15 


39-993 


1346.9 


30 


1191.8 


122.63 


2333-6 


40 


183.40 


2.934 


366 . 6 1 


40 


686.60 


40.992 


1363-4 


40 


1 200 . 5 


124.41 


2349.9 


50 


191-74 


3.207 


383-27 


50 


695.06 


42 . 004 


1380.0 


50 


I 209 . 2 


126. 20 


2366.2 


4° 


200 . 08 


3-492 


399-92 


ir 


703.51 


43-029 


1396.5 


24° 


1217.9 128.00 


2382.5 


10 


208.43 


3.790 


416.58 


10 


711.97 


44.066 


1413-I 


10 


1226.6 


129.82 


2398.8 


20 


216.77 


4.099 


433-24 


20 


720.44 


45.116 


1429.6 


20 


1235-3 


131.65 


2415.1 


30 


225.12 


4.421 


449.89 


30 


728.90 


46.178 


1446.2 


30 


1244.0 


133.50 


2431.4 


40 


233-47 


4.755 


466.54 


40 


737-37 


47.253 


1462.7 


40 


1252.8 


135-36 


2447.7 


50 
5° 


241 .81 


5. 100 


483.20 


50 


745-85 


48.341 


1479-2 


50 


1261.5 


137-23 


2464 . 


250.16 


5.459 499-85 


15° 


754-32 


49.441 


1495-7 


25° 


1270.2 


139.11 


2480.2 


10 


258.51 


5.829 516.50 


10 


762.80 


50.554 


1512.3 


10 


1279.0 


141 .01 


2496-5 


20 


266.86 


6. 211 533-15 


20 


771.29 


51.679 


1528.8 


20 


1287.7 


142.93 


2512.8 


30 


275.21 


6.606 549.80 


30 


779-77 


52.818 


1545-3 


30 


1296.5 


144.85 


2529.0 


40 


283.57 


7.013 566.44 


40 


788.26 


53-969 


1561.8 


40 


1305-3 


146.79 


2545-31 


50 
6° 


291 .92 


7.432 583-09 


50 


796.75 


55-132 


1578.3 


50 


1314.0 


148.75 


2561.5 


300.28 


7.863 599-73 


10° 


805.25 


56.309 


1594.8 


20° 


1322.8 


150.71 


2577.8 


10 


308 . 64 


8.307 616.38 


10 


813.75 


57-498 


1611.3 


10 


1331.6 


152.69 


2594.0 


20 


316.99 


8.762 


633.02 


20 


822.25 


58-699 


1627.8 


20 


1340.4 


154.69 


2610.3 


30 


325.35 


9-230 


649 . 66 


30 


830.76 


59-914 


1644.3 


30 


1349-2 


156.70 


2626.5 


40 


333.71 


9.710 


666 . 30 


40 


839.27 


61 . 141 


1660.8 


40 


1358.0 158.72 


2642.7 


50 


342.08 


10.202 


682.94 


50 


847-78 


62.381 


1677.3 


50 


1366.8 160.76 


2658.9 


7° 


350.44 


10.707 


699-57 


17° 


856.30 


63-634 


1693.8 


27° 


1375-6 


162.81 


2675.11 


10 


358.81 


11.224 


716.21 


10 


864.82 


64 . 900 


1710.3 


10 


1384-4 


. 164.87 


2691.3 


20 


367.17 


11-753 


732-84 


20 


873-35 


66.178 


1726.8 


20 


1393.2 


166.95 


2707.5! 


30 


375.54 


12.294 


749-47 


30 


881.88 


67.470 


1743-2 


30 


1402.0 


1 69 . 04 


2723.7 


40 


383.91 


12.847 


766. 10 


40 


890.41 


68.774 


1759.7 


40 


1410.9 


171.15 


2739-9 


50 


392.28 


13-413 


782.73 


50 


898.95 


70.091 


1776.2 


50 


1419.7 


173-27 


2756.1 


8° 


400 . 66 


13.991 


799-36 


18° 


907.49 


71.421 


1792.6 


28° 


1428.6 


175-41 


2772.3 


10 


409.03 


14.582 


815.99 


10 


916.03 


72.764 


1809.1 


10 


1437.4 


177-55 


2788.4 


20 


417.41 


15.184 


832.61 


20 


924.58 


74.119 


1825.5 


-20 


1446.3 


179.72 


2804.6 


30 


425.79 


15-799 


849-23 


30 


933-13 


Z5-488 


1842.0 


30 


1455. I 


181.89 


2820.7 


40 


434.17 


16.426 


865.85 


40 


941.69 


76.869 


1858.4 


40 


1464.0 


184.08 


2836.9' 


50 


442.55 


1 7 . 066 


882.47 


50 


950.25 


78.264 


1874.9 


50 


1472.9 


186.29 


2853.0 


0° 


450.93 


17.717 


899 . 09 


19° 


958.81 


79.671 


1891.3 


29° 


1481.8 


188.51 


2S69.2 


10 


459-32 


18.381 


915.70 


10 


967-38 


81 .092 


1907.8 


10 


1490.7 


190.74 


2885.31 


20 


467.71 


19.058 


932.31 


20 


975.96 


82.525 


1924.2 


20 


1499.6 


192.99 


2901 .4 


30 


476.10 


19.746 


948.92 


30 


984-53 


83.972 


1 940 . 6 


30 


1508.5 


195.25 


2917.6 


40 


484.49 


20.447 


965-53 


40 


993.12 


85-431 


1957.1 


40 


1517.4 


197-53 


2933-7 


50 


492.88 


21 . 161 


982 14 


50 
20° 


I 00 I .70 


86 . 904 


1973-5 


50 


1526.3 


199.82 


2949.8 


10° 


501 .28 


21.886 


998.74 


1010.29 


88.389 


1989.9 


30° 


1535.3 


202. 12 


2965.9 


10 


509.68 


22.624 


io^5-35 


10 


1018.89 


89.888 


2006 . 3 


10 


1544.2 


204.44 


2982.0 


20 


518.08 


23-375 


1031.95 


20 


1027.49 


91-399 


2022.7 


20 


1553-I 


206 . J7 


2998. 1 


30 


526.48 


24.138 


1048.54 


30 


1036.09 


92.924 


2039.1 


30 


1562. I 


209 . 1 2 


3014.2 


40 


534.89 


24.913 


1065. 14 


40 


1044.70 


94.462 


2055.5 


40 


1571.0 


211 .48' 


3030.2 


50 


543.29 


25.700 


1081.73 


50 
2V 


1053-31 


96.013 


2071 .9 


50 


1580.0 


213.86 


3046.3 


11° 


551-70 


26 . 500 


1098.33 


1061.93 


97-577 


2088.3 


31° 


T589.0 


216.25 3062.4 1 



318 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS FOR A 

1° CURVE. 



A Taiii^ent 


Kxt.Dist. 


LoiitfCh'il 


A 


Tan ire lit 


Kxt.Dist. LoiiirCh (1 


A 


Taiiireiit 


Kxt.nist. 


LonirChd 




T. 


i'. 


1 Lt. 




T. 


^. , Lt. 


T. 


A'. 


J.<. 1 


31" 


1 1589.0 


216.25 


3062.4 


41" 


2142.2 


387.38 4013.1 


51° 


2732.9 


618.39 


4933 4| 


lo'i 1598.0 


218.66 


3078.4 


10 


2151.7 


390.71 4028.7 


10 


2743.1 


622.81 


4948 


4 


20 


1 606 . 9 


221 .08 


3094.5 


20 


2161 .2 


394.06: 4044-3 


20 


2753.4 


627 . 24 


4963 


4 


30 


1615.9 


223.51 


3110.5 


30 


2170.8 


397-43; 4059 -9 


30 


2763.7 


631 .69 


4978 


4 


40 


1624.9 


225.96 


3126.6 


40 


2180.3 


400.82 


4075-5 


40 


2773.9 


636.16 


4993 


4 


50 


1633.9 


228.42 


3142.6 


50 


2189.9 


404.22 


4091 . I 


50 


2784.2 


640 . 66 


5008 


j4 
4 


32" 


1643.0 


230.90 


3158.6 


42^ 


2199.4 


407.64 


4106.6 


52° 


2794.5 


645.17 


5023 


10 


1652.0 


233.39 


3174.6 


10 


2209.0 


411.07 


4122. 2 


10 


2804.9 


649.70 


S038 


4 


20 


1661 .0 


235.90 


3 1 90 . 6 


20 


2218.6 


414.52 


4137-7 


20 


2815.2 


654.25 


S053 


4 


30 


1670.0 


238.43 


3206.6 


30 


2228. I 


417.99 


4153-3 


30 


2825.6 


658.83 


5068 




40 


1679.1 


240 . 96 


3222.6 


40 


2237.7 


421 .48 


4108.8 


40 


2835.9 


663.42 


5083 


3 


50 


1688. I 


243.52 


3238.6 


50 


2247.3 


424.98' 4184.3 


50 


2846.3 


668.03 


5098 


2 


33° 


1697.2 


246.08 


3254.6 


43° 


2257.0 


428. 50 ■ 4199.8 


53° 


2856.7 


672.66 


5113 


I 


10 


I 706 . 3 


248.66 


3270.6 


10 


2266.6 


432.04 42153 


10 


2867.1 


677.32 


5128 





20 


I715.3 


251 .26 


3286.6 


20 


2276.2 


435-59 4230.8 


20 


2877.5 


681 .99 


5142 


9 


30 


1724.4 


253.87 


3302.5 


30 


2285.9 


439-16; 4246.3 


30 


2S88.O 


686.68 


5157 


8 


40 


1733.5 


256.50 


3318.5 


40 


2295.6 


442.75 4261.8 


40 


2898.4 


69 I . 40 


5172 


7 


50 


1742.6 


259.14 


3334.4 


50 


2305.2 


446.35 4277.3 


50 


2908.9 


696 . 1 3 


5187 


6 


34° 


1751.7 


261 .80 


3350.4 


44^ 


2314-9 


449.98 4292.7 


54" 


2919.4 


700.89 


5202 


4 


10 


1760.8 


264.47 


3366.3 


10 


2324.6 


453.62 4308.2 


10 


2929.9 


705.66 


5217 


3 


20 


1770.0 


267. 16 


3382.2 


20 


2334.3 


457-27 43236 


20 


2940.4 


710.46 


5232 


I 


30 


1779. I 


269.86 


3398.2 


30 


2344.1 


460.95 4339-0 


30 


2951 .0 


715.28 


5246 


9 


40 


1788.2 


272.58 


3414.1 


40 


2353.8 


464.64! 4354-5 


40 


2961.5 


720. II 


5261 


7 


50 


1797.4 


275.31 


3430.0 


50 


2363.5 


468.35 4369-9 


50 


2972. I 
2982.7 


724.97 


5276 


5 


35" 


1806.6 


278.05 


3445-9 


45° 


2373-3 


472.08 4385-3 


55° 


729.85 


5291 




10 


1815.7 


280.82 


3461.8 


10 


2383-1 


475.82 


4400.7 


10 


2993.3 


734.76 


5306 


I 


20 


1824.9 


283.60 


3477.7 


20 


2392.8 


479.59 


4416.1 


20 


3003.9 


739.68 


5320 


9 


30 


1834. I 


286.39 


3493-5 


30 


2402 . 6 


483.37 


4431-4 


30 


3014.5 


744.62 


5335 


6 


40 


1843.3 


289.20 


3509-4 


40 


2412.4 


487.16 


4446 . 8 


40 


3025.2 


749.59 


5350 


4 


50 
30° 


1852.5 
1861.7 


292 .02 


3525-3 


50 
4(i° 


2422.3 


490 . 98 


4462 . 2 


50 


3035-8 


754-57 


5365 


. I 


294.86 


3541. I 


2432 . I 


494.82' 4477.5 


5(r 


3046.5 


759.58 


5379 


8 


10 


1870.9 


297.72 


3557.0 


10 


2441.9 


498.67 4492.8 


10 


3057.2 


764.61 


5,394 


S 


20 1880. I 


300.59 


3572.8 


20 


2451.8 


502.54 4508.2 


20 


3067 . 9 


769.66 


5409 


2 


30 


1889.4 


303.47 


3588.6 


30 


2461 .7 


506.42 4523- 5 


30 


3078.7 


774-73 


5423 


.9 


40 


1898.6 


306.37 


3604.5 


40 


2471.5 


510.33 4538.8 


40 


3089.4 


779.83 


5438 


.6 


50 


1907.9 


309.29 


3620.3 


50 


2481 .4 


514.25 4554-1 


50 


3100.2 


784.94! 5453 


3 


37° 


I917.I 


312.22 


3636. I 


17° 


2491.3 


518.20 4569.4 


57" 


3110.9 


790.08 5467 


9 


10 


1926.4 


315-17 


3651.9 


10 


2501 .2 


522.16 4584.7 


10 


3121.7 


795-241 5482 


S 


20 


1935.7 


318.13 


3667.7 


20 


2511 .2 


526.13 4599-9 


20 


3132.6 


800.42 


5497 


2 


30 


1945.0 


321. II 


3683.5 


30 


2521 . 1 


530.13 4615.2 


30 


3143-4 


805.62 


5511 


8 


40 


1954.3 


324.11 


3699.3 


40 


2531.1 


534.15 4630.4 


40 


3154-2 


810.85 


5526 


-4 


50 1963.6 


327.12 


3715.0 


50 


2541.0 


538.18 46457 


50 


3165.1 


816. 10 


5541 





38" 


1972.9 


330.15 


3730.8 


48° 


2551.0 


542.23 4660.9 


58" 


3176.0 


821.37 


5555 


6 


10 


1982.2 


333.19 


3746.5 


10 


2561 .0 


546.30 


4676. I 


10 


3186.9 


826.66 


SS70 


2 


20 


I99I.5 


336.25 


3762.3 


20 


2571.0 


550.39 4691.3 


20 


3197.8 


831.98 


5584 


7 


30 


2000 . 9 


339.32 


3778.0 


30 


2581 .0 


554.50 4706.5 


30 


3208.8 


S37.31 


5599 


3 


40 


2010.2 


342.41 


3793-8 


40 


2591.1 


558.63 4721.7 


40 


3219.7 


842.67 


5613 


8 


50 2019.6 


345.52 


3809.5 


50 

4y° 


2601 . I 


562.77, 4736.9 


50 


3230.7 


848.06 


5628 


3 

8 


39° 1 2029.0 


348.64 


3825.2 


2611 .2 


566.94 4752.1 


51)° 


3241-7 


853.46 5642 


10 2038.4 


351.78 


3840.9 


10 


2621 .2 


571.12 4767.3 


10 


3252.7 


858.89, 5657 


3 


20 


2047 . 8 


354.94 


3856.6 


20 


2631.3 


575.32 4782.4 


20 


3263.7 


£64.34; 5671 


8 


30 


2057.2 


358.11 


3872.3 


30 


2641,4 


579.54 


4797-5 


30 


3274-8 


869.82; 5686 


3 


40 


2066 . 6 


361 .29 


3888.0 


40 


2651.5 


583.78 


4812.7 


40 


3285.8 


875.32' 5700 


8 


50 2076.0 


364.50 


3903 - 6 


50 


2661.6 


588.04 
592.32 


4827.8 


50 


3296.9 


880.84 1 5715 


2 

7 


40 


2085.4 


367.72 


3919-3 


50° 


2671.8 


4842.9 


(>0" 


3308.0 


886.38; 5729 


10 


2094.9 


370.95 


3935-0 


10 


2681.9 


596.62 


4858.0 


10 


3319-1 


891.95 5744 


I 


20 


2104.3 


374.20 


3950.6 


20 


2692. I 


600.93 


4873-1 


20 


3330.3 


897.54 5758 


5 


30 


2113.8 


377.47 


3966.3 


30 


2702.3 


605.27 


4888.2 


30 


3341-4 


903. 15 1 5772 


9 


40 


2123.3 


380.76 


3981.9 


40 


2712. 5 


609.62 1 


4903 - 2 


40 


3352.6 


908.79; 5787 




50 


2132.7 


384.06 


3997 . 5 


50 
51° 


2722.7 


6 1 4 . 00 


4918.3 


^ 


3363-81 914-45, 


5801 


7 


41° : 2142.2 


387.38 


4013. I 


2732.9 


_6'8.39i 


4933.4 


«r 1 


3375 -oi 


920. 14 1 


5816. 






319 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS FOR A 

1° CURVE. 



A 
61° 

ID 
20 

30 
40 

50 


Tangent 
T. 


Ext.Dist. 


LongCh'd 


A 

71° 

10 
20 

30 
40 

50 


Tangent 
T. 


Ext.Dist. 


LongCird 
LC. 


A 


1 Tangent 


Ext.Dist. 
E. 


LongCh'd 
LC. 


3375-0 

3386.3 

3397.5 
3408 . 8 

3420.1 

3431.4 


920. 14 
925-85 
931-58 
937.34 
943.12 

948.92 


5816.0 
5830.4 
5844-7 
5859.1 

5873-4 
5887.7 


4086 . 9 
4099.5 
4II2.I 
4124.8 

4137-4 
4150. I 


1 308 . 2 

1315-5 
1322.9 

1330-3 
1337-7 
1345- I 


! 6654.4 
6668.0 
6681.6 
6695 • I 

6708 . 6 
6722. I 


81° 
10 
20 
30 
40 
50 


4893 - 6 

4908 . 

4922 . 5 
4937-0 

4951-5 

4966 . 1 


1805.3 
1814.7 
1824. I 

1833-6 
1843. I 
1852.6 


7442 . 2 

7454-9 
7467 - 5 
7480.2 

7492 - 8 
7505-4 


62° 

10 
20 

30 
40 

50 

68° 

10 
20 

30 

40 

50 


3442.7 
3454.1 
3465.4 
3476.8 
3488 . 2 
3499-7 


954.75 
960 . 60 

966.48 

972.39 
978.31 
984.27 


5902.0 
5916.3 

5930-5 
5944.8 

5959-0 

5973-3 


72= 

10 
20 

30 
40 

50 


4162.8 

4175-6 
4188.4 
4201 .2 
4214.0 
4226.8 


1352.6 
1360. I 
1367.6 
1375-2 
1382.8 

1390-4 


6735.6 
6749.1 
6762.5 
6776.0 

6789-4 
6802 . 8 


82° 
10 
20 
30 
40 
50 


4980.7 

4995-4 
5010.0 
5024.8 

5039-5 
5054-3 


1862.2 
1871.8 
1881.5 
189I .2 
1 900 . 9 
I9IO.7 


7518.0 

7530.5 

7543-1 
7555-6 
7568.2 
7580.7 


3511.1 
3522.6 

3534-1 
3545-6 
3557-2 
3568.7 


990.24 

996.24 
1002.3 
1008.3 
IOI4.4 
1020.5 


5987-5 
6001 .7 
6015.9 
6030 . 
6044 . 2 
6058.4 


73° 

10 

20 
30 
40 
50 


4239-7 
4252.6 
4265.6 
4278.5 
4291.5 
4304-6 


1398.0 

1405 . 7 

i4i3-5 
1421 .2 

1429.0 

1436.8 


6816.3 
6829.6 
6843 . 
6856.4 
6869.7 
6883.1 


83° 
10 
20 
30 
40 
50 


5069 . 2 

5084.0 

5099-0 

5113-9 
5128.9 

5143-9 


1920. 5 

1930-4 
1940.3 

1950-3 
I 960 . 2 
1970.3 


7593-2 
7605.6 
7618.1 
7630.5 
7643.0 
7655-4 


64° 

10 
20 

30 
40 

50 


3580.3 

3591-9 
3603.5 

3615-1 
3626.8 

3638.5 


1026.6 
1032.8 
1039.0 
1045.2 
I051.4 
1057.7 


6072. 5 
6086.6 
6100.7 
6114.8 
6128.9 
6143.0 


74° 

10 
20 

30 
40 
50 

75° 

10 
20 

30 
40 
50 


4317-6 
4330-7 

4343-8 

4356.9 
4370.1 

4383-3 

4396.5 
4409.8 

4423 . I 
4436.4 
4449-7 
4463 - 1 


1444.6 

1452.5 
1460.4 

1468.4 
1476.4 

1484.4 


6896.4 
6909.7 
6923.0 
6936 . 2 

6949-5 
6962 . 8 


84° 
10 
20 
30 
40 
50 


5159.0 

5174-1 
5189.3 

5204.4 
5219.7 

5234-9 


1980.4 
1990.5 

2000 . 6 
2010. 8 
2021 . I 
2031.4 


7667.8 
7680.1 
7692.5 

7704-9 

7717.2 

7729-5 


65° 

10 
20 

30 
40 

50 
66° 

10 
20 

30 
40 

50 


3650.2 
3661 .9 
3673.7 
3685.4 
3697 - 2 
3709.0 

3720.9 

673^.7 
3744.6 

3756.5 
3768.5 
3780.4 


1063.9 
1070.2 
1076.6 
1082.9 
1089.3 
1095.7 

1 102. 2 
IIO8.6 
III5.I 
II2I.7 
II28.2 
1134.8 


6157. I 
6171 . I 
6185.2 
6199.2 
6213.2 
6227.2 


1492-4 
1500.5 
1508.6 
1516.7 
1524.9 
1533-1 


6976.0 
6989.2 
7002 . 4 
7015.6 
7028.8 
7041.9 


85° 
10 
20 
30 
40 
50 


5250-3 
5265.6 
5281.0 
'5296.4 
5311-9 
5327-4 


2041.7 
2052. I 
2062 . 5 
2073.0 
2083.5 
2094 . I 


7741-8 

7754-1 
7766.3 
7778.6 
7790.8 
7803.0 


6241 .2 
6255.2 
6269. I 
6283.1 
6297.0 
6310.9 


76° 

10 
20 

30 
40 

50 


4476.5 
4489.9 

4503 -4 
4516.9 

4530.4 
4544-0 


1541.4 

1549-7 
1558.0 
1566.3 

1574-7 
1583. I 


7055 
7068 . 2 
7081.3 

7094-4 
7107.5 
7120.5 


86° 
10 
20 
30 
40 
50 


5343-0 
5358-6 
5374-2 

5389-9 
5405-6 

5421.4 


2 1 04 . 7 
2115.3 
2126.0 
2136.7 

2147.5 
2158.4 


7815.2 
7827.4 
7839.6 

7851.7 
7863.8 
7876.0 


67° 

10 
20 

30 
40 

50 
68° 
10 
20 

30 
40 

50 

6d° 

10 

20 

30 
40 

50 


3792.4 
3804.4 
3816.4 
3828.4 

3840.5 
3852.6 


II4I.4 
II48.O 

1154.7 
II61 .3 
II68.I 

ri74.8 


6324.8 

6338.7 
6352.6 
6366.4 
6380.3 
6394-1 


77° 
10 
20 

30 
40 

50 


4557-6 
4571.2 
4584-8 

4598-5 
4612.2 
4626 . 


1591.6 
1 600 . 1 
1608.6 
1617. I 
1625.7 
1634.4 


7133-6 
7146.6 

7159-6 
7172.6 
7185.6 
7198.6 


87° 
10 
20 
30 
40 
50 


5437-2 

5453-1 
5469.0 

5484-9 
5500.9 

5517-0 


2169. 2 
2180.2 
2191 . I 
2202.2 
2213.2 
2224.3 


7888.1 
7900.1 
7912.2 

7924.3 
7936.3 
7948 . 3 


3864.7 
3876.8 
3889.0 
3901 .2 

3913.4 
3925.6 


1181.6 
1188.4 
1195.2 
1202.0 
1208.9 
1215.8 ■ 

1222.7 
1229.7 
1236.7 

1243-7 
1250.8 
1257.9 


6408 . 
6421.8 

6435-6 
6449.4 

6463 . 1 
6476 . 9 


78° 
10 
20 

30 
40 

50 


4639.8 
4653-6 

4667.4 
4681.3 
4695 . 2 
4709.2 


1 643 . 
1651.7 
1 660 . 5 
1669.2 

1678. 1 
1686.9 


7211 .6 
7224.5 

7237-4 
7250.4 
7263.3 
7276.1 


88° 
10 
20 
30 
40 
50 


5533-1 
5549-2 

5565-4 
5581-6 

5597-8 
5614.2 


2235-5 
2246.7 

2258.0 
2269.3 
2280.6 
2292.0 


7960.3 

7972.3 
7984.2 
7996.2 
8008.1 
8020 . 


3937.9 
3950.2 

3962.5 
3974.8 
3987.2 
3999-5 


6490 . 6 

6504-4 
6518. I 
6531.8 

6545-5 
6559.1 


7D° 

10 
20 
30 
40 
50 


4723.2 
4737-2 
4751-2 
4765.3 
4779-4 
4793-6 


1695.8 
1704.7 

1713-7 

1722.7 

1731-7 
1740.8 


7289.0 
7301.9 

7314.7 
7327.5 
7340.3 
7353-1 


89° 
10 
20 
30 
40 
50 


5630.5 
5646.9 

5663.4 
5679-9 
5696.4 
57130 


2303.5 
2315.0 
2326.6 
2338.2 

2349-8 
2361.5 


8031 .9 
8043 . 8 

8055-7 
8067.5 
S079.3 
8091 .2 


70=: 
10 
20 

30 
40 

50 


4011.9} 

4024.4 

4036.8 

4049-3 
4061.8 

4074.4 


1265.0 
1272. I 

1279-3 
1286.5 

1293.7 

1300.9 


6572.8 
6586.4 
6600 . I 
6613.7 
6627.3 
6640 . 9 


80° 
10 
20 
30 
40 
50 


4808 . 7 
4822.0 
4836.2 

4850.5 
4864.8 
4879.2 


1749-9 
1759-0 
1768.2 

1777-4 
1796.0 


7365-9 
7378.7 
7391-4 
7404-1 
7416.8 

7429-5 


J)0° 

10 
20 
30 
40 
50 


5729-7 
5746.3 
5763-1 
5779-9 
5796-7 
5813.6 


2373.3 
2385.1 

2397.0 

2408 . 9 

2420.9 

2432.9 


8103.0 
8114.7 
8126.5 
8138.2 
8150.0 
8161.7 


71° 


4086 . 9 


1308.2 


6654.4 


§1° 1 


4893.61 


1805.3 


7442 . 2 


t)i° 


5830.5 


2444.9 


8173-4I 



320 



TABLE III.— SWITCH LEADS AND DISTANCES. 



LEAD-RAILS CIRCULAR THROUGHOUT; GAUGE 4' S|". See §262. 



Frog 

Number 

(«). 



4 
4-5 

5 

5.5 

6 

6.5 

7 

7-5 

8 

S.5 

9 

9-5 
10 

10.5 
II 

II. 5 
12 



Frog Angle (F) 



14 



12 40 
II 



15 00 

59 



10 23 
9 31 



25 16 
20 

38 



47 
10 



51 

16 



37 41 

09 10 

43 59 

21 35 

01 32 

43 29 

27 09 

12 18 

58 45 

46 19 



Lead (L) 
(Eq. 79). 



37-67 
42 -37 
47.08 

51.79 
56.50 
61.21 
65.92 
70.62 

75-33 
80.04 

84-75 
89.46 

94.17 

98.87 
103.58 
108.29 
113.00 



Chord (QT) 
(Eq- 77)- 



37-38 
42. 12 
46.85 
51-58 
56.30 
61.03 
65-75 
70.47 
75.19 
79.90 
84.62 
89.33 
94-05 
98.76 

103.47 
108.19 
112.90 



Radius of Lead 
Rails (r, Eq.7S). 



150.67 

190.69 
23542 
2S4.S5 
339.00 
397. S5 
461.42 
529.69 
602.67 
680.36 
762.75 

849-85 
941.67 
1038. 19 
1139.42 
1245.36 
1356.00 



Log r. 



2.I780I 
.2S032 

•371S3 
.45462 
.53020 
•59972 

. 6640<3 
.72402 
.78007 

-83273 

.8S238 

.92934 

2.97389 

3.01627 

. 05668 

.09529 

3.13226 



Degree of 
Curve ((/). 



38 46' 

30 24 

24 32 

20 13 

16 58 

14 26 

12 26 

10 50 

9 31 

8 26 

7 31 



Frog 
Number 



/ 

7.5 

8 

8.5 
9 



45 
05 
32 
02 
36 
14 



9 
10 
10 
II 
II 
12 



TURNOUTS WITH STRAIGHT POINT-RAILS AND STRAIGHT FROG-RAILS ; GAUGE 4' 8^'. See § 265. 



Frog 

Number 

in). 



4 

4-5 

5 

5.5 

6 

6-5 

7 

7-5 

8 

8.5 

9 

9.5 
10 

10.5 
II 

II-5 
12 



Switch 

PointAngle 

(a). 



40' 
40 

45 
45 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 



Length of 

Switch 

Point 



7-5 

7.5 

10. 

10. o 

15.0 
15.0 
15-0 



15 
15 
15 



15-0 



15 

15 



15.0 
15.0 
15-0 
15.0 



Length of 

Straight 

Frog-rail 

(7"). 



1.50 
1.69 
1.87 
2.06 
2.25 

2.44 
2.62 
2. Si 
3.00 

3.19 
3-37 
3-56 
3- 75 
3-94 
4.12 

4-31 
4.50 



Lead (L) 
(Eq. 90). 



32.20 
34.29 
41.85 
44-16 
56.00 
58.84 
61.65 
64.36 
67.04 
69.60 
72.20 
74.70 
77.04 
79.51 
81.82 
84.09 
86.16 



Chord 

(ST) 

(Eq. 88). 



23.09 
25-03 

29. ss 
32.03 
38.66 
41.34 
43.98 
46.50 

48. 99 
51. 38 
53.80 
56.11 
58.28 

60.57 
62.69 
64.78 
66.67 



Radius of 
Lead- 
rails 

(r.Eq.87). 



125.21 
159.25 
197-65 
240.44 
2S8.09 
340.19 
397-65 
460.00 

527.91 
600 . 94 
6S1.16 

767.11 

85S.14 

959-00 

1065.52 

II80. 16 

1299.93 



Log 



09764 
20208 

2958(3 
38100 

45953 
53172 
59950 
66276 
72256 
77883 
83325 
8S4S6 

93356 
98182 
02756 
07194 
11392 



Degree of 


Curve 


(^). 


47' 


05' 


36 


36 


29 


22 


24 


00 


19 


59 


16 


54 


14 


27 


12 


29 


10 


52 


9 


33 


8 


25 


7 


28 


6 


41 


5 


59 


5 


23 


4 


51 


4 


24 



Frog 

Number 

(«) 



4 
4-5 

5 

5.5 

6 

6.5 

7 

7-5 

8 

8.5 

9 

9-5 
10 
10.5 
II 

II-5 
12 



TRIGONOMETRICAL FUNCTIONS OF THE FROG ANGLES (F). 



Frog 
Number FrogAngle (F). 



4 
4-5 

5 

5.5 

6 

6.5 
7 



7. 

8 

S. 

9 

9- 
10 
10. 
II 
II . 
12 



5 



14 
12 
II 
10 

9 

8 
8 
7 
7 
6 
6 
6 
5 
5 
5 
4 
4 



15' 
40 

25 
23 
31 
47 
10 

37 
09 

43 
21 
01 
43 
27 
12 

58 
46 



00' 

49 
16 
20 
38 
51 
16 

41 
10 

59 
35 
32 
29 

09 

18 

45 
19 



Nat. sin F. Na 


t. COSi^. 


.24615 


96923 


•21951 j 


97561 


.19802 


98020 


.1S033 


9S360 


.16552 ! 


98621 


-15294 


98S23 


.14213 


98985 


•13274 


99II5 


.12452 


99222 


.11724 


99310 


.11077 


993S5 


.10497 


99448 


•09975 


99501 


.09502 


99548 


.09072 


995S8 


.08679 


99623 


•08319 


99653 



Log sin F. 



9.39120 

•34145 
. 29670 

.25606 

.21884 

•18453 

.1526s 

. 1230I 

.09522 

.06909 

.04442 

9.02107 

8. 9989 I 

.97781 

•95770 

.93S4S 

8. 92007 

321 



Log cos F. 



1.98642 
.98927 

•9913I 
.992S2 

•99397 
.99486 

•99557 
.99614 
. 9966(3 

• 99699 
•99732 
•99759 
.997S3 
.99803 
.99S20 
.99S36 
1.9SS49 



Log cot F. 



10.59522 
.647S2 
.69461 
-73675 
.77513 
.81033 

.8428s 

•87313 

•9013s 

.92790 

.9528C3 

.97652 

IO.99S92 

11.02021 

.04050 

.05987 

II.07S42 



Log vers .F. 



8.48811 
.38721 
.29670 
.21467 
. 13966 

•0705s 

S. 0065 5 

7.94691 

.8911(3 

.83S64 

.78915 

.74232 

.6978S 

•65560 

.6152S 

.57676 

7.539S6 



Frog 

Number 



4 

4.5 

5 



5- 

6 

6. 

7 

7 ■ 

8 

8. 

9 

9- 
10 
10. 
II 
II . 
12 



5 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 





1 














"0 





d 


LO 


t^ 


Cl 


LO 


In 


























CO 


M 


^ 


CO 


"^ 


M 





CO 







O O 0\ ^-r\ xj-\ \0 O 





N 


CO 






© 






















5i 


O O 0\ On CO VO ro 
O O CN a\ ON ON On 


oo 


tN 
IN 


NO 






00 


n 


LO 


CO 
CO 







VO 


CO 


tN 

CO 







lA 6 '^ C> ^ On '^ 


ON 


-^ 


On 






























C) LO t\ On f^ "^ t^ 


On 


n 


':+ 



































N 


CNl 








^ 


'^ 


CO 


CO 


CO 


M Cl 


'"' 


^ 










"i-O 

CO 






CO 
IN 


tN 



00 


LO 


c^ 

LO 
^ LO 


t\ 

CO 

LO 


LO 





C 




CO 
IN 




too ("~» (— <0 <^* <f^ "-• 


"^ 


■^ 








Ci 


NO 


H 


vO vO oo CO O ^ r^ 




NO 


S 








CO 


o 





'^ 


Cl 


CO 





CO 





CO 


>-^ "^ — ri vO -^ — 


LTl 




O 




























ic 




-^ 


On 


ri 





























o 


•^ >-< Ln On " ro LO 


t^ 


OO 


q 








CO 


CO 


CO 


Cl 


Cl 


d 1— 


>^ 











od d\ d\ d\ 6 6 6 


d 


d 


>-<' 




; 




































>1 


LO 


tN 





Cl 


LO r^ 





Q 


LO 


Cl 
















X 






CO 

LO 


CO 

t^ 


LO 

NO 


'^ 

CO CO 









CO 


Cl 

ON 




























LO 


^ 


CI 





'^ 


d LO 


CO 





CO 







^N. vo n oo o — r^ 


_ 


M 


ON 









CI 


CI 


n 


Cl 




h-i 













H 


rj m CO -• o oo -• 
O — to oo >-n -^ 00 


NO 


ON 


OO 




























LO 


IN 


"^ 




1 
1 




b 


IN 


LO 


CJ 





IN LO 








Cl 


LO 


d d d d -< cj fo 


LO 


JN 


d 
























-^ 


LO 


CO 


CO H- 








LO 


" 














C 


l> 


LO 


CO 




00 


CO 

1— c 


w 


NO 
10 Cl 












CO 







VO 
CO 


-e- 


("^ (00 -^ CO On 1- (n 
LO t^ oo "-» vo On r^ 


(04 


M 


LO 

In 




a 

3 
1-1 




°(N 


M 


X 


UN n;^ 




_C 


























VO O M vo 00 O O 


CO 


M 


NO 






^ 






















« 


t^ ro m i^ M rj t^ 


CO 


OO 


LO 








r^ 





CI 


LO 


r-^ 








LO 


Cl 





In 


> 


m CO On CO t^ o ^' 


'^ 


NO 


00 






m 





LO 


HH 





CO 





1— 1 


Cl 








lt\ 






































O 


rt vr^ vA ^O "O t^ t^ 


IN. 


IN. 


tN 




c 





^ 






















N^ 


















NO 


1— 


CO 


Cl 





NO 


-^ 


LO 


CO 












• 

ft 


-a 
"5. 

3 
o 
(J 







CO 










Cl 








Cl 




LO 




Cl 


LO 


X 


On (^ (On (vO ('^ On vo 

O C^ON0O r^>-r\H-^ 

O OnOnOOnOnOnoo 


(Q <NO 
On CO 

tN NO 




*Lo 


CJ 

LO 


8 


CO 


LO 


8 




CO 


Cl 

LO 


LO 


tN 








w 


O On On On On On On 


ON 


On 


o\ 


o 


























© 


O On On 0\ On On On 


On 


ON 


ON 


4i 

OP 


'o 
a 


L*: 


- 







LO 


LO 

CO 


00 






Cl 


VO 


CO 


CO 


LO 


d d\ On On On d\ On 


On 


On 


d\ 












<M 







































10 


0) 




~ 


'-' 




















'-' 


'-' 


M 












*^ 


































W 


rt 




"Vi 


LO 


IN 








LO Cl 





In 


LO 


n 


-e- 


(CO (0 (M (^^ O ("^ <r\ 


<'+ 


O 


<0 




*-> 




LO 


^ 











'^ Cl 


CO 





w 


LO 




r^ ON On t^ oo oo NO 


NO 


CO 




c 
c 
























s 


00 LO vo oo "^ O >-o 


"^ 




CO 


'^ 


^ 




















X 


CO -1 w CO - NO CO 


ON 


On 


IN 


ft 


NO 


00 


CO 


LO 





CO On 


Cl 


CO 


VO 


VO 


tc 


CO oo -' CO "^ NO t^ 


CO 


0\ 


o 




rt 




-^ 


CO 


Cl 


^ 





-H CO 





Cl 


LO 


Cl 


o 




00 


co" 


On 


b 

o 

o 


V 

































S 
o 






















^ 


" 


^ 


n 




"0 


IN 


LO 








n LO 


IN 





CJ 


LO 








■■■■^ 


















CO 


CO 


l-l 








vo hm 





CO 


Cl 


'^ 


■e- 












M 
























X 

© 


., ., On (r^ (-^ (On (« 
3^ 2^ On On On oo 00 
C C On <JN On ON ON 


On 

NO 

On 


LO 

ON 


CO 
C) 

ON 




c 
.2 ■ 

u 


CO 


"tn 

C^ 





- 


8 


LO 


CO LO 


CO 


IN 

CO 


? 


CO 
CO 


cS 


O O On ON On On On 


On 


ON 


On 



























K, 


H- 


Cl 


Cl 


<<?; 












Q 






























































In 








LO 


c^ 


IN 


LO 


Cl 





In 
















^1 



"co 


CO 
IN 












00 


CO 


LO 


CO 
tN 


CO 

LO 


-e- 
























I— 








"^ 


M 


■^ LO 




5- 





CO 


ss 


(N (vn « oo t^ CO (0 


(-+ 


o 


IN 




























X 


r' NO CO — n LTi — 
O O — c) CO '"t- NO 


CO 
In 


OO 

On 


On 


























« 


M 


Cl 


Cl 


•<-i 


q q q q q q o 


q 


q 


►- 






























V 








CJ 


LO 


t\ 


Cl 


LO 


In 





^ 
























w 


"co 






CO 


LO 

NO 


CO 


c 

CO 


Cl 

On 




CO 
LO 


CO 
Cl 












4^ 


i 






















Cl 


-^ 


t-< 


^ 





CO 




o o o o o o o 


o 


o 


O 




























tc 


CO CO O O CO CO O 


o 


CO 


CO 





























Si ^ 


"K, D VO iJ-» C) t~^ O 


o 


IN 


M 























« 


'-' 


1— 1 


c< 


M 


t^-^ 

M ^ 




^ 




















n -^ -- Lo CO CO 


CO 


CO 


LO 








b 


LO 


c< 





IN 


LO Cl 





tN 


LO 







^ 
















-^ 


M 


CO 





►-. LO 





CO 


Tt- 


Cl 


z 


O O O i-i "-I ci CO 


'^ 


LO 


NO 






G? 






















u 














b 


CO 


<:> 


t^ 


CO 


- VO 


LO 


LO 


00 


^ 































Cl 




•^ LO 




^ 


CO 


LO 


Cl 
Cl 












-4^ 


t/j 
























s 


>-t C^ ro -^ "-> vo t^ 


CO 


On 


o 




























p 












*-Cj 


































■55 


rt 


©^ 


fH 


^1 


cc 


rj^ 


Lt C 


b» 0(0 


a 






322 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



s 






as 0\ o\ o\ oo r^rt-o "* 

n -^ r^ On c~» Tf t^ C\ r> "+ 

H^ „ ►H -I c^ cs 






vO^O M ro^ »-nr> >-oo O 
r^-±DO M ^ \D oo O •- CO 



i-r^mroooooor-^ri fO 

-< -c n 



<0 -t CO (ro ro -+ CO <— <rO <^ 
vOcoco i-nvo r~^cocovO f^ 

ONON"~>CAfOOOO O C) t1- 
tJ- lA \d o t^ r^ i^ 00* CO CO 



-e- 


iO\ 


(vO 


i-r> 


<CXD 


rv, (t^ 


\-r\ 


ri 


<r^ 


t^ 




0\ 


On 


CO 


iJ-> 


O - 


r\ 


NO 


ur> 


ro 


s 

V 


ON 


OS 


On 


ON 


On CO 


NO 


-t- 


>-( 


r^ 


ON 


ON 


C7n 


On 


ON ON 


ON 


CA 


ON 


CO 


if) 


ON 


ON 


On 


ON 


ON ON 


ON 


ON 


ON 


UN 


e 


ON 


ON 


ON 


ON 


ON On 


ON 


ON 


ON 


ON 



(— (D 


rj CO 


o 


rn 


fON 


ro 


<m 


<o 


CO On 


On no 


NO 


-+ 


CO 


ro 


ri 


o 


ON vO 


r-^ On 


LO 




LO 


-+ 


O 


NO 


m « 


— ro 


»M 


NO 


CO 


On 


ON 


r^ 


VO « 


-* NO 


CO 


ON 


o 




M 


m 



t^OOCOCOCOCO OnOnOnOn 



^, ON <0 <0 <CO CO OJ^ t^ CO (r<-) 

"-' ONONONr^i-nn j^O — 

- OnOnOnOnOnOncoco r\ 

O OnOnOnOnOnOnOnCnOn 



(CO — n NO 'i- "^ (CO <-+ '- r^ 

—frONO rO"^" >-iNO «-/^r^ 
O — ^^ -+ NO ON r> i/^ On f") 
OOOOOO-'-'-rJ 



i-r»w-iO O Lou-iO O i-nvn 
— •+rom^— 00«-t 



O O ^ <^l rO"^r^CN« rO 



-< M cO'^h^^NO t>^co OnO 



C3 

I 



Ci 



OC 



C 



^ 



CO 



*1 



o^ 



CI lo On M l^ Q i-*"! O 'o O 



NO "^ CO r^ « O ro n NO "^i O 
»'^ n -rt O <■! ro ro ro ri ►- I O 



CO CO r^ t>x NO "^ 



ro r> ►- I O 



CO -t- O — CO 



r-^ CO i'-^ NO n 

■-I -rt ^ ro LO 



to 



rn o 
o - 



yr\ O 



- o 



rv NO NO »-^ "+ Tf c<-) n — 



r>. O Lo 

■<^ CO •- 



00 ri — 



O >-o 

O -* 



i/~i ro 



O u^ 

CO — 



r\ NO 



u-iioT^-'^fOri — •- 



8 


8 


O 
O 


8 


- 






O >-o 

CO Th 



r^ CO 
O " 



O "-> O 
O — CO 



O NO r^ 
ro O CO 



V? 8 



ro *-<"> 

O r) 



rf Tt CO CO N 



lo O 

-. ro 




o 




8 


« 






O ir^ O 

O -r CO 



O ro n 

o o - 



"^ 8 



— O CO 
CI O CO 



O "^ 

ro — 



Cl NO 

O ci 



CO CO M CI M 



O »J~i O >-o 
ro '^ O - 



M oo O NO 
LO Tj- l-O LO 



O >- CJ CO 



O Lo O 
CO -^ O 



c) CO O 
C) O '^ 



ir-i 


o 

CO 


o 
o 


- 


CO 


c 
o 


-- 


O 






lr^ O •'N O 
'i- CO - O 



CO r^ NO O 
ro ci ci ro 



LO o ""> 

'^ CO — 



CO t^ VO 
ro — >-^ 



« « O O 



O u-i 

ro -rj- 



O io O ro 
O - ro -t 



r\ CO 

ro — 



"^ NO CI ro 
O iJ^i lo *-<-> 



O >-■ ci Cl CO -^ 



o 


lO 


o 

CO 


8 


LO 


^ 




o 
o 


D 

o 








o 



i-r\ O 

-t CO 



ro C> 
O -i- 



'S 8 



uo oo 

•rt CJ 



NO ^^ CO r\ 
ci ►- O O 



O NH «-l C> CO ""^ «J^ 





8 


o 
o 




NO 


VO 


o 
o 




o 

o 


O 


o 





O tr\ 

ro -1- 



ri CO 
Cl -t 



o - 



O NO 

Cl "~> 



O w-i CO «- 
ro •+ •-'N '-' 



r^ ro -f — 
ro C) — " 



GO — — cico-t-LO 



o 

CO 


8 


O 


8 


°o 






^ CO 



iT 8 



CO f^ 
CO VO 



NO O 

Cl O 



vi-i O CO vr-i 
Tt CO — LO 



CO Cl — -t 
ro Cl — O 



OOO '-'<'» «^'fO-t"~» 



O >o 

ro -t 



r-^ CO 

O " 



O vn 

O - 



\r\ VO 
ro "-o 



O "^ 

ro -f 



Cl ro 
Cl "^ 



O CO NO oo 

O H- Cl CO 



O ►- t^ oo 

ro — 1-0 -+ 



OOOO^^''^'^''^-*' 



^ ^ ^1 W ^ It C t- X Ct O 



323 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 





















*l-n 


fS 








(S NO 


































C) 


-* 





fs 


-^ 





CO 


CO 










55 


O OS r^ ^ 
O ON On On 


NO 


On O 
w-i On 

'd- CO 


On 


NO ■* 
r^ CO 
CO o 









LO 




LO 


XI 
CO 


LO 








s- 


LO C) 
LTl 




CO 










LTi CN "^ On 


^ 


On CO 


r^ 


•-<* Tt- 








r^ 





ir% 


rj- 


n 


h. 


On 


t^ Tf 


M 









M '^ r^ On 


r) 


1? J^ 


On 














'"' 




'"' 


'"' 














"0 


n 


U-l 


ON 


LO 





































CO 


\_r\ 




CO 



LO 


CO 






CO 






























un 


t>~. 





("1 


r^ 






(■xt- (O CO <CO 


(n 


On (T^ 


r^ 


r^ CO 






CO 


CO 


CO 


-' 


^ 





M 


n ^ 





CO 




H 


rv» t-^ oo NO 


NO 


r^ O 


CO 00 "- 1 































rv. NO fo '+ 


r^ 


LO n 


'* 


r^ NO 








•^ 


CO 





1—1 


ON 00 





-^ M 





r) 




^ 

NJ 


m CO CO -- 


r-^ 


On OO 


-^ 


oo — 








1—1 


•-1 


1— 1 


►™ 
















d\ 6\ 6 6 


d 


ON I-. 

d «■ 


CO 


"I" ^. 
































"uo 





^ 


-^ 









































QO 




l-O 


CO 






CO 



10 


CO 
n 











LO 


CO 


































CO 


-^ 


-^ 


CO 





CO 


10 








'-' 


CO 






On 1^ t-^ i-H 


M 


CO CO 


On 


p< r^ 














0\ 00 


rv. 


LO 


CO 


n 





n 


^ 






O "^ n t^ 


ON 


O O 


On 


VO .- 








1— 1 


—1 






















S 


d 6 «-< CO 


ON 


On M 
On lo 


q 


r^ CO 

d •-<' 




j^ 






























i 
































N 


CO -^ 




to 




uo 


T^ 


M 















































CO 





CO 





CO 











CO 























b» 


8 


On 


ri 


LO 




NO 





CO 











8 


n 

rl- NO 




-6- 














B 

3 




00 


rv. 


-^ 




NO 00 t^ On (M 
NO 00 00 m rj 


On rj 
OO CO 


(ON 

NO 


O (CO 

LO O 




1/) 






























^ 






















« 


O -^ NO O 
OO CO CO CO 


CO 


CO C) 


On 

CO 

NO 


- CO 

oo LO 
CO O 








CO 






CO 


8 




CO 


8 


8 


CO 


8 




CO 




© 


Lo NO rC. t>^ 


tv. 


oo' oo' 


oo' 


oo' On 




c 





^ 






















H^ 














<u 


n 





r^ 


10 


M 








Lo r~^ 





n 
















^^ 


J3 

-a 






NO 




NO 








LO 


CO 






"^ CO 
1- CO 


^ 
tn 


LO 






























cS 


■5. 




























•.^ 






















O 


(00 Lo <o <"^ 

On OO ■^ CO 
On On On OO 


NO 


(no (O 
NO On 

D NO 


(0 

oo 


(NH CO 

NO ON 




3 

y 
u 






^0 




CO 


8 



CO 


8 


8 






CO 




CO 






V 


On On On On 


On 


<7n CO 


t^ 


NO "^ 


as 


























O 
1^ 


On On On On 


ON 


ON On 


ON 


ON On 




c 


10 


In 


r^ 





M 


ITN 








t-^ "^ 


N 


On 




Cf\ On CTn On 


d\ 


On d\ 


On 


On d\ 


0) 


'0 
a 




"^ 





M 


n 


1-1 





CO 


"~i 


u^ 


LO 

























'=!■ 


CO 


r< 


t— 





- 


CO Tl- NO 


00 




















^0 











c 











un 


00 




-6- 


rf n O <On 


o 


(CO (fv. 


oo 


^ (O 


c 




CO 





CO 











CO 


CO 


w^ 


^ 




_ 


OO On OO M 


r^ 


NO NO 


On 


CO -^ 


u 

1 

o 


V 


























in 


O r^ CO 
a\ rt- r>^ On 


LO 


O CO 

NO CO 
C< CO 


ON 

oo 


00 NO 

Lr> \0 


be 

B 


^ 





CO 




8 






UO 


CO 


c^ 








o 


l< Oo' OO' CO 


On 


d\ On 


Cfv 


ds dv 


V 





































N 


e 





CO 


c^ 


'-' 


^ 





'"' 


M 


"i^ VO 


t^ 









"0 




















NO 


^ 



































u 







CO 











CO 





CO LO 


n 


'* 




-0- 














c 


CO 


"d 


N 


\-r\ 








t^ 


VO 


M On 


t^ 


^ 




s 


(On (NO <no rJ 


t-^ 


(N CO 


(O 


On O 







LO 


n 


Tt 











n 


LO M 








o 


On On OO NO 


N^ 


CO O 




CO r^ 




«^ 


























On ON ON On 


On CO r\ 


\^r\ 


n oo 






























«J 


On On On On 


On 


0\ <JN 


On 


On oo 




<c 




M 


HH 








M 


ri 


CO 


-^ NO 


00 







eS 


• • • • 


• 


• " 


• 


• " 




(U 


























?^ 














Q 




























^ 








































'0 




















00 Tj- 


li^ 


8 




















ei 


ro 












LO 


CO 





CO 


LO C) 


ON 


















•e- 


















ir\ 


CO 





•* 


CO 


Tf 


LO 


N- t:^ 


n 


M 




— 


(t^ M (CO (i-« 


O-O 


(N On 


O 


t^ (r^ 






























'S 


CO NO M r^ 

O n Lo 00 


O n -' 
CO OO ^ 


On 



n 1-1 

oo nO 








°o 











'-' 


r< 


CO 


lJ-> NO 


00 









o q q q 


"• 


i-c n 


CO 


CO '^t 
































\> 

















On NO 


00 


rv 




















^ 













CO 



l-O 


CO 


10 

ON 


n Lo 





ON 


















« 
















^ 





CO 





uo 


LO 


LO 


~ '^ 


n 







s 


^O o o 
CO CO O O 


o 

CO 


O O 
CO O 


O 

O 


O O 
CO CO 


















« 


K4 


M 


CO 


ir\ \o 


CO 







sr 


O l-l CO LO 


t^ 


O -+ OO 


n r^ 


















































00 


CO LO 





■LTt 










t— 1 HH 


■"" 


ci n 














CO 





CO 





M 


W) « 


CO 


CO 


















0» 


'0 


u-v 


t^ 





n 


10 


t^ 


On f^ 


'* NO 





















c 







CO 






LO 




CO 


rf NO 




CO 

On 




© 














M 
























•-" N CO rt 


ir\ 


NO t^ 


00 


ON O 




c 


































^^ 








0» 


H 


CI 


W 


Tj^ 


10 





t' X 


a 





















rt 




















1H 





324 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 


1 


1" 12 1 :5 


4 5 , a 1 


7 S ! 1 


1*. P. 1 




100 

lOI 

I02 


00 000 

. 432 
860 


043 


087 


130 


173 216 


260 


303 


775 
*i99 


389 
817 

^241 
^7 ^ • 






475 
902 


518 
945 


561 
987 


C04 
*o3o 


646 689 
^072 *ii4 


732 
*i57 


43 

[ 4.5 


43 

4.3 


42 

4.2 


41 

41 




103 
104 


01 283 
703 


326 
745 


368 
787 


410 
828 


452 
870 


494 
911 


53t> 
953 


578 
994 


619 
^036 


661 .. 

*o77 


2 8.7 
3 130 


8.6 
12.9 


8.4 
12.6 

rA 


8.3 
13 3 

ifi A 1 




105 


02 119 


160 


201 


243 


284 


325 


3^6 


407 


448 


489 : 


4 »74 

5 21.7 


17.2 
21. s 


21.0 


20.5 




106 


530 


571 


612 


653 


694 


735 


775 


816 


857 


898 


6 26. 1 


25.8 


25-2 


24.6, 




107 
108 


938 
03 342 


979 

382 


^019 

422 


*o6o 
463 


* 
^100 

503 


*i4i 

543 


*i8i 

583 


*22I 
623 


*262 
663 


*302 

703 ; 


7 304 

834.8 

(^i3y-i 


30.1 
34-4 
38.7 


29.4 
33.6 
37.8 


28.7 
32.8 

36-» 




109 

110 

III 

112 


742 
04 139 


782 


822 


862 


901 


941 


981 


*026 


*o6o 


*ioo 






178 


218 


257 


297 


33^^ 


375 


415 


454 


493 

883 

^269 




532 
922 


57? 
960 


616 
999 


649 

*o38 


688 
*o76 


727 
'^115 


766 
*i54 


805 

*I92 


844 

*23I 


40 

I 4.0 


40 

4.0 


39 

3-9 


38 

3.8I 




113 


05 308 


346 


384 


423 


461 


499 


538 


576 


614 


652 


2 8.1 

3 12. i 


8.0 
12.0 


7.8 
11.7 


7.6 
II. 4 




114 


690 


728 


765 


804 


842 


880 


9^8 


956 


994 


*032 






ir, 6 


It;. 2 




115 


06 070 


107 


145 


183 


220 


258 


296 


333 


371 


408 . 


5 20.2 


20.0 


'9-5 


19.0 




116 


446 


483 


520- 


558 


595 


632 


670 


707 


744 


781 • 


624.3 


24.0 


23.4 


33.8 




117 


818 


855 


893 


930 


967 


*oo4 


*04o 


*o77 


*i 14 


*i5i • 


728.3 
832.4 


28.0 
32.0 


27 -3 
31.2 


36.6 
30-4 




118 


07 188 


225 


261 


298 


335 


372 


408 


445 


481 


518 . 


9 36.4 


36.0 


351 


34.2 




119 

120 

121 


554 


591 


627 


664 


700 


737 


773 


809 
*i76 

529 


845 


882 


3f 77 76 ■?«; 




918 


954 


990 


*026 

386 


*o62 


*098 


*i34 


*206 

564 


*242 




08 278 


314 


350 


422 


457 


493 


606 




122 


636 


671 


707 


742 


778 


813 


849 


884 


920 


955 


I 3-7 


3-7 


3.6 ^.i;| 




123 

124 


995 
09 342 


*026 

377 


*o6i 
412 


*o96 
447 


*i3i 

482 


*i66 
517 


*202 

552 


■*"237 
586 


*272 
621 


*307 
656 


2 7-5 

3 II. 2 


7-4 
11. 1 


7.2 
10.8 


7.0 
10.5 




I2S 


691 


725 


760 


795 


830 


864 


899 


933 


968 


*002 


4 150 

5 18.7 


14.8 

18.5 


14.4 
18.0 


14.0 
17-5 




126 


10 037 


071 


106 


140 


174 


209 


243 


277 


312 


346 


6 22.5 


22.2 


21.6 


21.0 




127 


380 


414 


448 


483 


517 


551 


585 


619 


^53 


687 


7 26.2 

8 ^0.0 


25.9 

2Q.6 


25.2 

?a 8 


24-5 

28.0 




128 


721 


755 


789 


822 


^56 


890 


924 


958 


991 


*025 


9 33-7 


33-3 


324 


31-5 




129 

130 

131 
132 

133 


II 059 


092 


126 


160 


193 


227 


260 


294 


327 


361 






394 


427 


461 


494 


528 


561 


594 


627 


661 


694 




727 
12 057 

385 


760 
096 
418 


793 
123 

450 


826 
156 
483 


859 
189 

515 


892 

221 

548 


925 

254 
580 


958 

287 

613 


991 

320 

645 


*024 

352 
678 


34 

1 3-4 

2 6.9 


34 

3-4 
6.8 


33 

3-3 
6.6 


32 

3-3 

6.4 




134 


7x5 


743 


775 


807 


840 


872 


904 


937 


969 


*OOI 


3 10.3 


10.2 


9.9 


9.6 




135 


13 033 


065 


097 


130 


162 


194 


226 


258 


290 


322 


4 138 

5 172 


13.6 
17.0 


13.2 
16. s 


12.8 

16.0 




136 


354 


386 


417 


449 


481 


513 


545 


577 


608 


640 


6 20.7 


20.4 


19.8 


19.2 




137 

138 


672 

988 


703 
*oi9 


735 
*o5i 


767 

*o82 


798 
*ii3 


830 
*i45 


862 
*i76 


893 

*207 


925 
*239 


956 

^270 


7 24.1 
827.6 
931.5 


23.8 
27.2 
30.6 


23.1 
26.4 
29.7 


22.4 

25.6 

28.8 




139 
110 

141 


14 301 


332 


364 


395 


426 


457 
767 

*o75 


488 


519 


550 


582 


31 -?! ^0 20 




613 


644 


675 


706 


736 


798 
*io6 


829 


866 891 1 




922 


952 


983 


*oi4 


*o45 


*i37 


*i67 


^198 




142 


15 229 


259 


290 


320 


351 


381 


412 


442 


473 


503 


1 3.1 


3.1 


30 


3.9 




143 

144 


533 
836 


564 
866 


594 
896 


624 
926 


655 
956 


685 
987 


715 
*oi7 


745 
*o47 


770 
*o77 


806 
*io7 


2 6.3 

3 9-4 

4 12.6 

5 «5-7 


6.2 
9-3 


6.0 
9.0 


5.8 
8.7 

TT ^ 




14s 


16 137 


165 


196 


226 


256 


286 


316 


346 


376 


405 


12.4 
15-5 


15.0 


14-5 




146 


435 


465 


494 


524 


554 


584 


613 


643 


672 


702 


6 18.9 


18.6 


18.0 


17-4 




147 


731 


761 


791 


820 


849 


879 


908 


938 


967 


997 


7 22.0 

8 25.2 


21.7 

74 R 


31. 
24.0 


20.3 
23.3 




148 


17 026 


055 


085 


114 


143 


T72 


202 


231 


266 


289 


928.3 


27.9 


27.0 


a6.i 




T49 
150 


318 
609 


348 


377 


406 


435 


464 


493 


522 


551 


580 






638 


667 


696 


725 


753 


782 


811 


840 


869 




N. 





1 1 2 


3 4 


5 


1 6 1 7 


8 


9 


P.P. 





325 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 
150 

151 
152 
153 
154 
155 
156 

157 
158 
159 
160 

161 
162 
163 

164 

165 
166 

167 
168 
169 

170 

171 
172 

173 

174 

175 
176 

177 
178 
179 
180 

181 
182 

183 
184 

185 
186 

187 
188 
189 

190 

191 
192 

193 
194 

195 
196 

197 
198 
199 

200 

N. 



O 



17 609 



897 

18 184 
469 

752 

19 °33 

312 

590 
865 

20 139 



2 



412 



682 

951 

21 219 

484 

748 

22 on 

271 
53T 
788 



23 045 



299 

553 
804 

24 055 
304 

551 

797 

25 042 

285 



527 



768 

26 007 

245 
482 
717 

951 

27 184 
416 
646 



875 



28 103 
330 
555 
780 

29 003 
225 

446 
666 

885. 

3Q 103 

O 



638 



926 
213 

497 
780 
061 

340 
617 

893 
167 



439 



709 

978 
245 

511 
774 
037 

297 
557 
814 



070 



667 



955 

241 

526 

808 
089 
368 

645 
926 

194 



792 
031 
269 

505 
740 

974 

207 

439 
669 



898 



466 



736 



126 
352 

578 
802 
025 
248 

468 
688 

907 
124 



350 
603 

855 
105 

353 
600 

846 
091 

334 



575 



816 

055 
292 

529 
764 

998 

230 

462 

692 



921 



696 



984 

270 

554 

836 
117 

396 

673 
948 
221 



493 



763 



'012 

298 

582 

864 

145 
423 
700 

975 
249 



520 



790 



•032 f^o58 



149 

375 
600 

825 
048 
270 

490 
710 

929 

146 



298 

564 

827 

089 

349 
608 
865 



121 



375 
628 
880 

129 

378 
625 

871 

115 

358 



599 



325 
590 
853 
115 

375 
634 
891 



147 



401 

653 

905 

154 

403 
650 

895 

139 

382 



840 

078 
316 

552 

787 

'021 

254 
485 
715 



944 



171 

398 
623 

847 
070 

292 

512 
732 
950 
168 



623 



863 
102 
340 

576 
811 

'044 

277 
508 

738 



966 



194 
426 

645 
869 
092 

3M 

534 
754 
972 
190 



753 



'041 

327 
611 

893 
173 

451 

728 

^003 

276 



6 



782 



811 



547 



817 

^'085 

352 

616 
880 
141 

401 
660 
917 



'070 

355 
639 
921 
201 

479 

755 
^030 

303 



574 



172 



426 
679 

930 
179 

427 
674 

920 
164 

406 



647 



007 

126 

599 

834 

*o68 

300 

531 
761 



989 



844 
'112 

378 

643 
906 

167 

427 
686 

942 



'098 
384 
667 

949 
229 

507 

783 

'057 

33^ 



8 



9 



840 



601 



198 



451 

704 

955 

204 

452 
699 

944 
188 

430 



672 



911 

150 
387 
623 
858 
'''091 

323 

554 
784 



012 



239 
465 
696 

914 

137 

358 

578 

798 

*oi6 

233 
6 



871 
'139 
405 
669 
932 
193 

453 
711 

968 



*'l27 

412 
695 

977 
256 
534 
8i5 
*o85 
357 



869 



*i56 

446 
724 
*oo5 
284 
562 

838 
112 

385 



628 



223 



477 
729 
980 

229 
477 
723 

968 

212 

455 



696 



898 
^165 

43 J 

695 
958 
219 

479 
737 
994 



249 



502 

754 
'005 

254 
502 

748 

993 

237 
479 



935 

174 

411 

646 

881 

= 114 

346 

577 
806 



•035 



262 
488 
713 

936 

159 

380 

6o5 

820 

*o38 

254 



720 



959 

197 

434 

670 

904 

'137 

369 
600 

829 



058 



285 
510 
735 

959 
181 

402 

622 
841 

'059 

276 

8 



^5 
924 
^^192 
458 
722 
984 
245 

505 

763 

^019 



P. P. 



274 



527 

779 

'030 

279 

526 

773 

''017 

261 

503 



744 



983 
221 

458 

693 
928 

*i6i 

392 
623 

_85l 
*o8o 



307 
533 
758 
981 
203 
424 
644 
863 
»o8i 
298 
9 



29 28 27 



. I 


2.9 


2.8 


2. 


.2 

■3 


5-8 
8.7 


5.6 
8.4 


5- 
8. 


■4 


11.6 


T1.2 


10. 


•5 
.6 


M-5 
17.4 


14.0 
16.8 


13- 
16. 


•7 


20.3 


ig.6 


18. 


.8 


23.2 


22.4 


21. 


•9 


26.1 


25.2 


24. 



2§ 26 



. I 


2-6 


.2 


5-3 


•3 


7-9 


•4 
• S 


TO. 6 

13.2 


.6 


15-9 


■7 


18.5 


.8 


21.2 


•9 


23-8 



2.6 

5-2 

7.8 

10.4 
13.0 
15-6 



20.8 
234 



2S 25 24 



. I 


2.5 


2. K 


.2 


5-1 


50 


•3 


7-6 


7-5 


•4 


10.2 


TO.O 


•5 


12.7 


12.5 


.6 


15-3 


150 


■7 


17-8 


17-5 


.8 


20.4 


20.0 


•9 


22.9 


22.5 



2.4 

4.8 

7.2 

9.6 

12.0 

14.4 
16.8 

19. a 
21 .6 



. I 


2-3 


.2 

•3 


4-7 
7.0 


4 


9-4 


•5 
.6 


II. 7 
14. 1 


•7 
.8 

•9 


16.4 
18.8 
21. 1 



23 23 

2-3 

4.6 
6.9 

9.2 

11-5 
13-8 

16. 1 
18.4 
20.7 



22 22 21 

2.1 

4-3 
6.4 

8.6 

TO. 7 
12.9 

15.0 
17.2 
19-3 



.1 


2.2 


2.2 


.2 

•3 


6.7 


4.4 

6.6 


•4 


9.0 


8.8 


•s 


II. 2 


II. 


.6 


13-5 


13.2 


•7 
.8 


15-7 
18.0 


15-4 
17.6 


•9 


20.2 


19.8 



P. p. 



326 









TABLE V.- 


—LOG 


ARITinL^ ( 


)F XI 


'MHERS. 








Tn. 





1 


2" 


3 4 


5 


7 S 


<) 


r. p. 1 


200 


30 103 


124 


146 


168 190 


211 


233 


254 


276 


298 






j 20I 


319 


341 


3^3 


384 


406 


427 


449 


470 


492 


513 


. I 


2 2 


2 I 


202 


535 


556 


578 


599 


621 


642 


664 


685 


707 


728 





4-4 


4.2 


203 


749 


771 


792 


813 835 


856 


878 


899 


920 


941 


• 3 


6.6 


6.3 


204 


963 


984 


♦005 


*o27 *o48 


^069 


*og6 


*II2 


^^^33 


*i54 




R R 


R 1 


205 


31 175 


196 


217 


239 


260 


281 


302 


323 


344 


365 


•4 

.5 


II .0 


0.4 
10.5 


206 


386 


408 


429 


450 


471 


492 


513 


534 


555 


576 


.6 


13.2 


12.6 


207 


597 


618 


639 


660 


681 


702 


722 


743 


764 


785 








208 


806 


827 


848 


869 


890 


910 


931 


952 


973 


994 


•7 
.8 


154 
17.6 


14.7 

16.8 


209 

210 


32 014 

222 


035 


056 


077 


097 


118 


139 


160 


186 
387 


201 
407 


•9 


19. s 


18.9 


242 


263 


284 


304 


325 


346 


3^6 


35 -1 


211 


428 


449 


469 


490 


510 


531 


551 


572 


592 


613 




20 


^KJ 


212 


^33 


654 


674 


695 


715 


736 


756 


776 


797 


817 


.2 


4. 1 


4.0 


213 


838 


858 


878 


899 


919 


940 


960 


980 1*001 


*02I 


•3 


6.1 


6.0 


214 


33 041 


061 


082 


102 


122 


142 


163 


183 j 203 


223 






8.0 ! 
10. t 


215 


244 


264 


284 


304 


324 


344 


365 


385 405 


425 


• 4 
•5 
.6 


0.2 
10. 2 


216 


44! 


465 


485 


505 


525 


546 


566 


586 606 


626 


12.3 


12.0 


217 


646 


666 


6S6 


706 


726 


746 


766 


786 806 


825 






L 


218 


845 


865 


885 


905 


925 


945 


965 


985 *oo4 *024 


•V 
8 


14-3 
16.4. 


14.0 
16.0 


219 
220 

221 


34 044 


064 


084 


104 


123 


M3 


163 


183 203 222 


•9 


1S.4 


18.0 ; 


242 
439 


262 


281 


301 1 321 


341 


366 


380 400 I 419 


19 19 


459 


478 


498 


5^8 


537 


557 j 576 


596 , 615 


222 


^35 


655 


674 


694 


713 


733 


752 I 772 


791 i 811 


. I 
. 2 


1.9 
3-9 

5-8 


1.9 

3.8 
5-7 


223 


830 


850 


869 


889 


908 


928 


947 1 9^6 986 


*oo5 


•3 


224 


35 025 


044 


063 


083 


102 


121 


141 166 179 


199 








225 


218 


237 


257 


276 


29! 


3''4 


334 


353 372 


391 


•4 


7.8 

9-7 
II. 7 


7.6 

9-5 
II. 4 


226 


411 


430 


449 


468 


487 


S<^7 


526 


545 


564 


583 


• 5 
.6 


227 


602 


621 


641 


660 


679 


698 


717 


736 


755 774 








228 


793 


812 


83? 


850 


869 


88h 


907 


926 


945 9^4 


•V 

.8 

•9 


13-6 
15-6 
17-5 


13-3 


229 
230 

231 


983 


*002 


"^021 


*o4o 


*o59 


*o78 


*o97 1*116 1*135 ;*i54 


15-2 

17. 1 


36 173 


191 


216 


229 


248 


267 


286 1 305 


323 ' 342 


„ 1 


361 


380 


399 


417 


436 


455 


474 


492 


511 


530 


18 


10 


232 


549 


567 


586 


605 


623 


642 


661 


679 


698 


717 


. I 

2 


1-8 
3-7 

5-5 


1 .8 ! 

3-6 ; 

5-4 1 


233 


735 


754 


773 


791 


810 


828 


847 


866 


884 


903 


■ 3 


|234 


921 


940 


958 


977 


996 


*oi4 


*o33 


*o5i 


*o7o 


*o88 








1 235 


37 107 


125 


143 


162 


186 


199 


217 


236 


254 1 273 


•4 


7-4 

9.2 

II .1 


7.2 
9.0 

10.8 


I236 


291 


309 


328 


346 


364 


383 


401 


420 


438 


456 


• 5 
.6 


237 


475 


493 


511 


530 


548 


5^6 


584 603 


621 


639 








238 


657 


676 


694 


712 


730 


749 


767 785 


803 


821 


•7 

.8 

•9 


12.(3 

14.8 
16.6 


12.6 


1239 
240 

241 


840 


858 


876 


894 


912 


930 
III 


948 i 967 


985 1*003 


14.4 
16.2 


38 021 


039 


057 


075 


093 


129 ) 147 


165 183 




201 


219 


237 


255 


273 


291 


309 


327 


345 


3^3 




17 


1 242 


38i 


399 


417 


435 


453 


471 


489 


507 


525 


543 


. I 



1-7 

3-5 

5-2 


1.7 

^•4 
5-1 


'243 


566 


578 


596 


614 


632 


650 


667 685 


703 


721 


•3 


'244 


739 


757 


774 


792 


810 


828 


845 863 


881 


899 








|245 
1 246 


9^6 
39 093 


934 
III 


952 
129 


970 
146 


987 
164 


*oo5 
181 


*023 

199 


*046 
217 


*o58 
234 


*o76 

252 


•4 

•5 
.6 


7.0 

8.7 
10.5 


6.8 ; 

8.5 1 

10.2 


1247 


269 


287 


305 


322 


340 


357 


375 


392 


410 


427 








I248 


445 


462 


480 


497 515 


532 


550 


567 


585 


602 


.7 

.8 

•9 


12.2 


II. 9 
13.6 


! 249 
250 


620 


637 


655 


672 689 


707 


724 


742 


759 


776 


14 -O 

157 


794 


811 


828 


846 863 


881 


898 


915 


933 


950 




i N. 

1 





1 


2 


3 4 


5 : 7 1 8 1 9 


P. P. 1 



327 











TABLE V.- 


-LOGARITHMS < 


3F NUMBERS. 










1 N. 

250 

251 





1 ! 2 


3 4 


5 


6 


7 1 8 





p. p. 




39 794 


811 


828 


846 


863 


881 


898 


915 


933 


950 








967 


984 


*002 


*oi9 


*o36 


*o54 


*o7i 


*o88 


*io5 


*I23 




252 


40 140 


157 


174 


191 


209 


226 


243 


266 


277 


295 




1 




253 


312 


329 


346 


z^i 


380 


398 


415 


432 


449 


466 




if 


17 




254 


483 


500 


517 


534 


551 


569 


586 


603 


620 


637 


. I 


1-7 


1.7 




255 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 


.3 


3-5 
5.2 


3-4 

5.1 1 




255 


824 


841 


858 


875 


892 


908 


925 


942 


959 


976 










257 


993 


*OIO 


^027 


*044 


*o6i 


*o77 


*094 


*iii 


*I28 


""145 


•4 


7.0 

R ^ 


6.8 

8.5 

10.2 




25« 


41 162 


179 


195 


212 


229 


246 


263 


279 


296 


3^3 


•5 
.6 


0.7 
10.5 




259 
260 

261 


32,^ 


346 


3^3 


380 


397 


413 


430 


447 


464 
631 


480 
647 


•7 

.8 


12.2 
14.0 

15.7 


II. 9 
13.6 
15. -^ 




497 
664 


514 


536 


547 


564 


581 


597 


614 




680 


697 


714 


730 


747 


764 


780 


797 


8n 




262 


830 


846 


863 


880 


896 


913 


929 


946 


962 


979 








263 


995 


*OI2 


•^023 


*o45 


*o6i 


^078 


^094 


*iii 


*I27 


*i44 








264 


42 160 


177 


193 


209 


226 


242 


259 


275 


292 


308 




- 5 ^ 




265 


324 


341 


357 


373 


390 


406 


423 


439 


455 


472 




16 


10 




266 


488 


504 


521 


537 


553 


569 


586 


602 


613 


635 


.1 




1-6 
3-3 
4.9 


1.6 




267 


651 


667 


6^% 


700 


716 


732 


748 


765 


781 


797 


■3 


4.8 




268 


813 


829 


846 


862 


878 


894 


910 


927 


943 


959 










269 
270 

271 


975 


991 


^^007 


*o23 *o4o 


^056 


^072 


*o88 


*io4 


*I26 


•4 

•5 
.6 


6.6 
8.2 
9.9 


6.4 
8.0 
9.6 




43 ^36 


152 


168 


184 


200 


216 


233 


249 


265 


281 




297 


3^Z 


329 


345 


361 


377 


393 


409 


425 


441 




272 


457 


473 


489 


505 


520 


536 


552 


568 


584 


600 


• / 

8 


II-5 
13.2 

14-8 


11 .2 

12 8 




273 


6ig 


632 


648 


664 


680 


695 


711 


727 


743 


759 


•9 


14.4 




274 


775 


791 


8og 


822 


^Z^^ 


854 


870 


SS6 


901 


917 






i 


275 


933 


949 


965 


980 


996 


*OI2 


*028 


*043 


*oS9 


*o7s 






) 


1 276 


44091 


105 


122 


138 


154 


169 


185 


201 


216 


232 






\ 


277 


248 


263 


279 


295 


310 


326 


342 


357 


373 


38Q 




iS 


15 


i 


278 


404 


420 


435 


451 


467 


482 


498 


513 


529 


545 


2 


1-5 
3-1 
4-6 

6.2 

7-7 


1-5 
3-0 

4-5 

6.0 

7-5 




279 
280 

281 


560 


576 


591 


607 


622 


638 


653 


669 


685 


700 


.3 
.4 


i 


716 
870 


731 


747 


762 


778 


793 


809 


824 


839 


855 
*oo9 




886 


901 


917 


932 


948 


963 


978 


994 




282 


45 025 


040 


055 


071 


085 


102 


117 


132 


148 


163 


.6 


9-3 


9.0 




283 


178 


194 


209 


224 


240 


255 


270 


286 


301 


3^6 




^ 


[ 




284 


332 


347 


362 


377 


393 


408 


423 


438 


454 


469 


.7 
.8 


10. 8 
12.4 


10.5 1 
12.0 




2«5 


484 


499 


515 


530 


545 


560 


576 


591 


606 


621 


•9 


13-9 


13-5 




286 


^2>6 


652 


667 


682 


697 


712 


727 


743 


758 


773 








287 


788 


803 


818 


^ZZ 


848 


864 


879 


894 


909 


924 








288 


939 


954 


969 


984 


999 


*oi4 


*029 


*044 


*o59 


*o75 




^ 




289 
290 

291 


46 090 


105 


120 


135 


150 


165 


180 


195 


210 


225 


. I 

.2 
•3 


14 

1.4 

4-3 


^4 

1-4 

2.8 
4.2 




240 


255 


269 


284 


299 


314 


329 


344 


359 


374 




389 


404 


419 


434 


449 


464 


479 


493 


508 


523 




292 


538 


553 


568 


583 


597 


612 


627 


642 


657 


672 


.4 


5.8 
7.2 


5.6 
7.0 




293 


687 


701 


716 


731 


746 


761 


775 


790 


805 


820 




294 


834 


849 


864 


879 


894 


908 


923 


938 


952 


967 


.6 


8.7 


8.4 




295 


982 


997 


'^OII 


^025 


^41 


*o55 


*o75 


*o85 


*IOO 


*ii4 


.7 

.8 


ICt T 


n R 




296 


47 129 


144 


158 


173 


188 


202 


217 


232 


246 


261 


II. 6 


9.0 
II. 2 




297 


275 


290 


305 


319 


334 


348 


3^3 


378 


392 


407 


•9 


13.0 


12.6 




298 


421 


436 


451 


465 


480 


494 


509 


523 


538 


552 








299 
300 

N. 


567 


58I 


596 


610 


625 


639 654 


66s 


683 


697 








712 


726 


741 


755 


770 


784 799 


813 


828 


842 







1 


2 


3 


4 


5 1 6 


7 


8 9 


P. P. || 



328 



TABLE v.— T.OGARITTIMS OF yUMT^FRq. 



300 

301 


1 


2 


3 


4 


5 





7 


8 


J) 


r. i». 1 


47 712 


726 


741 


755 


770 


784 


799 


813 


828 


842 




856 


871 


885 


900 


914 


928 


943 


957 


972 


986 


302 


48 000 


015 


029 


044 


058 


072 


087 


101 


11? 


130 




3<'3 


144 


158 


173 


187 


201 


216 


230 


244 


259 


273 




304 


287 


301 


316 


33^ 


344 


358 


*^ -^ 

0/0 


387 


401 


415 




30s 


430 


444 


458 


472 


487 


50i 


515 


529 


543 


SS8 


« 


306 


572 


586 


606 


614 


629 


^43 


657 


671 


685 


699 




^'f 


*^ 


307 


714 


728 


742 


756 


770 


784 


798 


812 


827 


841 


2 


1 .4 

2.9 


1 .4 

2.S 


308 


855 


869 


^^3 


897 


911 


925 


939 


953 


967 


982 


•3 


4-3 


4.2 


309 
310 

311 


996 'OIO 


*024 


*o38 


*052 


=^^066 


*o8o 


*o94 


*io8 


*I22 


•4 

• 5 

.6 


5.S 
7.2 

8.7 


5.6 
7.0 , 
8.4 


49 136 i 150 


164 


178 


192 


206 


220 


234 


248 


262 


276 


290 


304 


318 


332 


346 


359 


373 


387 


401 


312 


415 


429 


443 


457 


471 


485 


499 


513 


526 


540 


•7 
.8 




9.8 , 
II. 2 1 


3^3 


554 


5^8 


582 


596 


610 


624 


^37 


65 J 


665 


679 


II. 6 


314 


693 


707 


720 


734 


748 


762 


776 


789 


803 


817 


•9 


13.0 


12.6 


315 


831 


845 


858 


872 


886 


900 


913 


927 


941 


955 




316 


9^8 


982 


996' 


*OIO 


=^023 


*o37 


*o5i 


*o65 


*078 


*092 




317 


50 106 


119 


^33 


147 


160 


174 


188 


201 


215 


229 




31S 


242 


256 


270 


283 


297 


311 


324 


33^ 


352 


365 




319 
320 

321 


379 
515 


392 


406 


420 


433 
569 


447 


466 


474 


488 


soil 


t5 to 


528 


542 


555 


583 


596 


610 


623 1 637 


650 


664 


677 


691 


704 


718 


731 


745 


758 772 


, I 


-0 

1-3 

2.7 


'^ 1 

1-3 
2.6 


322 


785 


799 


812 


826 


839 


853 


866 


880 


893 


907 


.2 


323 


920 


933 


947 


960 


974 


987 


*OOI 


*oi4 


*02 7 


*04i 


•3 


4.0 


3-9 


324 


51 054 


068 


081 


094 


108 


121 


135 


148 


161 


175 


•4 


5.4 
6.7 


5-2 
6.5 


325 


188 


201 


215 


228 


242 


255 


268 


282 


295 


308 


326 


322 


335 


348 


361 


375 


388 


401 


415 


428 


441 


.6 


8.1 


7.8 


327 


455 


468 


481 


494 


508 


521 


534 


547 


561 


574 


•7 
.8 


9-4 
10.8 


r> T 


328 


587 


6o5 


614 


627 


640 


653 


667 


680 


693 


706 


y .1 
10.4 


329 
330 

33^ 


719 


733 


746 


759 


772 
904 


785 


798 


812 


825 


838 


•9 


12. i 


II. 7 


851 
983 


864 


877 


891 


917 


930 


943 


956 


969 




996 


*oo9 


*022 


*o35 


*o48 


*o6i 


*o74 


*o87 


*ioo 


332 


52 114 


127 


140 


153 


166 


179 


192 


205 


218 


231 






244 


257 


270 


283 


296 


309 


322 


335 


348 


361 




334 


374 


387 


400 


413 


426 


439 


452 


465 


478 


491 




335 


504 


517 


530 


543 


556 


569 


582 


595 


608 


621 


r S 1 '> 


33^ 


634 


647 


660 


672 


685 


^98 


711 


724 


737 


750 




I 2 


I 2 


337 


763 


776 


789 


801 


814 


827 


840 


853 


866 


879 


. 2 


2.5 


2.4 


338 


891 


904 


917 


930 


943 


956 


9^8 


981 


994 


*oo7 


.3 


3.7 


3.6 


339 


53 020 


033 


045 


058 


071 


084 


097 


109 


122 


135 






/I R 


340 

341 


148 


i65 


173 


186 


199 


211 


224 


237 


250 
377 


262 


•4 

•5 
.6 


5 -^ 

6.2 

7.5 


4.0 
6.0 
7.2 


275 


288 


301 


313 


326 


339 


352 


364 


390 


342 


402 


415 


428 


440 


453 


466 


478 


491 


504 


516 


. 7 


S.7 
10. 


8.4 
9.6 


343 


529 


542 


554 


567 


580 


592 


605 


618 


630 


643 


.8 


344 


656 


66s 


681 


693 


706 


719 


731 


744 


756 


76(3 


•9 


II. 2 


10.8 j 


345 


782 


794 


807 


819 


832 


845 


857 


870 


882 


895 




346 


907 


920 


932 


945 


958 


970 


983 


995 


*oo8 


*020 




347 


54033 


045 


058 


076 


083 


095 


108 


120 


^33 


14! 




348 


158 


170 


183 


195 


208 


220 


232 


245 


257 


270 




349 
350 


282 


295 


307 


320 


332 


344 


357 


369 


3S2 


394 




407 


419 


431 


444 


456 


469 


481 


493 


506 


5^S 


N. 





1 


2 


1 3 


4 


»■> 


1 


7 1 8 


9 


P.P. 



329 









TABLE v.- 


-LOGARITHMS ( 


3F NUMBERS. 






1 N. 





1 


2 j 3 


4 


5 6 


7 8 9 1 


P. 


P. 


350 

351 


54 407 


419 


431 


444. 


456 


469 


481 


493 


506 


518 


. I 


12 

1 . 2 


530 


543 


555 


568 


580 


592 


605 


617 


629 


642 


\352 


654 


666 


679 


691 


703 


716 


728 


740 


753 


765 


.2 


2.5 


353 


777 


790 


802 


814 


826 


839 


851 


863 


876 


888 


•3 


3-7 


1354 


900 


912 


925 


937 


949 


961 


974 


986 


998 


*oi6 


.J. 


K .0 


i355 


55 023 


035 


047 


059 


071 


084 


096 


108 


120 


133 


.5 


6.2 


1356 


145 


157 


169 


181 


194 


206 


218 


230 


242 


254 


.6 


7.5 


1357 


267 


279 


291 


303 


315 


327 


340 


352 


364 


376 


.7 
.8 


8.7 

10. 


1358 


3H 


400 


412 


424 


437 


449 


461 


473 


485 


497 


359 
360 

1361 


509 
630 

750 


521 


533 


545 


558 


570 
696 


582 


594 


606 


618 


•9 


II. 2 

12 

I 2 


642 


654 

775 


666 


678 


702 


714 


726 


738 


762 


787 


799 


811 


823 


835 


847 


859 


3^2 


871 


883 


895 


907 


919 


931 


943 


955 


966 


978 


. 2 


2.4 


\3^3 


990 


*002 


*oi4 


^025 


"^038 


*o56 


^062 


*o74 


*o86 


^098 


•3 


3.6 


364 


56 no 


122 


134 


146 


158 


170 


181 


193 


205 


217 




4.8 
6.0 


365 


229 


241 


253 


265 


277 


288 


306 


312 


324 


33^ 


•4 

.5 


366 


348 


360 


372 


383 


395 


407 


419 


431 


443 


455 


.6 


7.2 


367 


465 


478 


490 


502 


514 


525 


537 


549 


561 


573 




8.4 
9.6 

10.8 
II 


368 


585 


596 


6o§ 


620 


632 


643 


655 


667 


679 


691 


•7 
.8 


369 
370 

371 


702 


714 


726 


738 


749 


761 
879 


773 


785 


796 


808 


•9 


820 


832 


843 855 


867 
984 


896 1 902 


914 


925 


937 


949 


961 


972 


996 


*oo7 


*oi9 


*o3i 


^042 


372 


57 054 


066 


077 


089 


lOI 


112 


124 


136 


147 


159 


.2 


2.3 


I373 


171 


182 


194 


206 


217 


229 


240 


252 


264 


275 


•3 


3-4 


374 


287 


299 


310 


322 


333 


345 


357 


3^S 


380 


391 




4-6 

"^ . 7 


J375 


403 


414 


426 


438 


449 


461 


472 


484 


495 


507 


•4 


376 


519 


530 


542 


553 


565 


576 


588 


599 


611 


622 


.6 


6.9 


377 


634 


645 


657 


668 


680 


691 


703 


714 


726 


737 




8.0 

9.2 

10.3 

II 


378 


749 


760 


772 


783 


795 


806 


818 


829 


841 


852 


•7 
8 


379 
380 

381 


864 


875 


887 


898 


909 


921 


932 


944 


955 


967 


•9 


978 


990 


*OOI 


*OI2 *024 


*o35 
149 


*o47 


*o58 


^'069 


*o8i 


58 092 


104 


115 


126 


138 


161 


172 


183 


195 


382 


206 


217 


229 


240 


252 


263 


274 


286 


297 


3^S 


.2 


2.2 


3^3 


320 


33^ 


342 


354 


365 


376 


388 


399 


410 


422 


•3 


3-3 


384 


433 


444 


455 


467 


478 


489 


501 


jI2 


523 


535 






385 


546 


557 


568 


580 


591 


602 


613 


625 


636 


647 


•4 

•5 
.6 


4.4 

5.5 
6.6 


386 


658 


670 


681 


692 


703 


715 


726 


737 


748 


760 


387 


771 


782 


793 


804 


816 


827 


838 


849 


861 


872 






388 


883 


894 


905. 


916 


928 


939 


950 


961 


972 


984 


•7 

Q 


7-7 
8.8 

9 9 
10 


389 
1390 

391 


995 
59 106 


*oo6 


*oi7 


*028 


*o39 


*o5o 
162 


*o62 


*o73 
184 


^084 


*o95 


•9 


117 


128 


140 


^51 


173 


195 


206 


217 


229 


240 


251 


262 


273 


284 


295 


3^6 


317 


392 


328 


339 


35^ 


362 


373 


384 


395 


406 


417 


428 


. I 


I.O 


1393 


439 


450 


461 


472 


483 


494 


505 


5I6 


527 


53S 


.3 


3.1 


394 


549 


560 


571 


582 


593 


604 


615 


626 


637 


648 






395 


659 


670 


681 


692 


703 


714 


725 


736 


747 


758 


•4 


4.2 


396 


769 


786 


791 


802 


813 


824 


835 


846 


857 


868 


•5 
.6 


5-2 
6.3 


397 


879 


890 


901 


912 


923 


933 


944 


955 


9^6 


^977 






398 


988 


999 


*OIO 


*02I 


*032 


*o43 


*o53 


*o64 


*o75 


*o86 


.7 


7-3 
8.4 
9.4 


399 
400 


60 097 
206 


108 


119 


130 


141 


151 


162 


173 


184 


195 


.8 
.9 


217 


227 


238 


249 


260 


271 


282 


293 


3^3 





1 


2 


3 


4 


5 6 


7 


8 9 1 


P. 


P. 



330 









TABLE V.- 


-LOGARITHMS ( 


)F NUMHERS. 






In. 

400 

401 


1 


'21314 


5 e 


7 1 8 





I* 


. V. 


60 206 


217 


1 227 1 238 


249 


260 


271 
379 


282 


293 
401 


412 






314 


325 


1 zz(^ 


347 


357 


z^l 


390 


402 


422 


433 


444 


455 


466 


476 


487 


498 


509 


519 






403 


530 


541 


552 


563 


573 


584 


595 


606 


616 


627 




II 


,404 


638 


649 


659 


670 


681 


692 


702 


713 


724 


735 


. I 
.2 


I . I 
2.2 


405 


745 


756 


767 


777 


788 


799 


810 


826 


H^ 


842 


•3 


, 3-3 


406 


852 


863 


874 


884 


895 


906 


916 


927 


938 


949 






407 


959 


970 


981 


991 


*00 2 


*oi3 


*023 


*o34 


*044 


*o55 


.4 

.5 


4.4 

5-5 


408 


61 066 


076 


087 


098 


I08 


119 


130 


140 


151 


161 


.6 


6.6 


409 
410 

411 


172 


183 


193 


204 


! 215 


225 
331 


236 

342 


246 
352 


257 


268 


.7 

.8 

•9 


7.7 
S.8 
9.9 


278 ' 289 


299 1 310 


326 


Ti^Z 


373 


384 1 394 


405 


416 


426 


437 


447 


458 


468 


479 


412 


489 


500 


511 


521 


532 


542 


553 


563 


574 


584 






413 


595 


605 


616 


625 


637 


647 


658 


668 


679 


689 






414 


700 


710 


721 


731 


742 


752 


763 


773 


784 


794 






415 


805 


8'5 


825 


%l6 846 


857 


867 


878 


888 


899 






416 


909 


Q20 


930 


940 


951 


961 


972 


982 


993 *oo3 


.2 


2. 1 


[417 


62 013 024 


034 


045 


^y:i 


065 


076 


086 


097 


107 


•3 


3-1 


418 


117 


128 


^zl 


149 


159 


169 


180 


190 


200 


211 






419 
420 

421 


221 

325 
428 


232 


242 


252 


263 

3^6 
469 


273 

376 

480 


283 


294 


304 


314 


•4 
•5 
.6 

•7 

.8 


4.2 

5-2 

6.3 

7-3 
8.4 


335 


345 


356 
459 


387 


397 


407 418 


438 


449 


490 


506 


510 521 


422 


531 


541 


552 


562 


572 


582 


593 


603 


6^s 


624 


423 


634 


644 


654 


665 


675 


685 


695 


706 


716 


726 


.9 


9.4 


424 


736 


747 


757 


767 


777 


7S8 


798 


808 


818 


828 






425 


839 


849 


859 


869 


879 


890 


900 


910 


926 


931 






426 


941 


951 


961 


971 


981 


992 


*002 


*OI2 


*022 


*032 






427 


63 043 


053 


063 


073 


083 


093 


104 


114 


124 


134 




10 


428 


144 


154 


164 


175 


185 


195 


205 


215 


225 


235 


. I 

.2 


I.O 

2.0 


429 
430 

431 


245 
347 


256 


266 


276 


286 

387 


296 


3O6 


3I6 


326 


336 


•3 

•4 

.5 


3-0 

4.0 
5-0 


357 


367 1 377 


397 407 


417 


427 


437 


447 


458 


468 


478 


488 


498 


508 


518 


528 


538 


432 


548 


558 


5^8 


578 


588 


598 608 


618 


628 


639 


.6 


6.0 


433 


649 


659 


669 


679 


689 


699 709 


719 


729 


739 






434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 


•7 
.8 


7.0 
8.0 


435 


849 


859 


869 


879 


889 


899 


909 


919 


928 


938 


•9 


9.0 


436 


948 


958 


9^^8 


978 


988 


998 


'■^^oog 


*oi8 


*028 


^038 






437 


64 048 


058 


068 


078 


088 


09S 


107 


117 


127 


137 






43^ 


147 


157 


167 


177 


187 


197 


207 


217 


226 


236 






439 
440 

441 


246 


256 


266 


276 


286 


296 
394 


306 


315 


325 


335 


.1 
.2 

.3 


9 

0.9 J 

^•9 

2-8 


345 
444 


355 
453 


365 


375 


384 


404 


414 


424 


434 


463 


473 


483 


493 


503 


512 


522 


532 


442 


542 


552 


562 


571 


58i 


591 


601 


611 


621 


636 






443 


640 


650 


660 


670 


679 


689 


699 


709 


7I8 


728 


•4 

.5 
.6 


3-s : 

4-7 

5.7 


444 


738 


748 


758 


767 


777 


787 


797 


806 


8I6 


826 


445 


836 


846 


855 


865 


875 


885 894 


904 


914 


923 






446 


933 


943 


953 


962 


972 


982 


992 


*OOI 


*oii 


*02I 


.7 
.8 


6.6 
7.6 


447 


65 031 


040 


050 


060 


069 


079 


089 


098 


108 


118 


•9 


8.5 1 


448 


128 


137 


147 


157 


166 


176 


186 


195 


205 


215 






449 
450 


224 


234 


244 


253 


263 


273 


282 


292 


302 


311 






321 


7^7>^ 


1 340 


350 ! 360 


3^^9 379 


389 


398 


408 


i ^^• 





1 


1 2 1 3 1 4 


r> is 


7 1 8 ! 


V 


. P. 



331 









TABLE V.- 


-LOGARITHMS 


OF NUMBERS. 


4 




I~x7 

450 

451 


1 1 j 2 


3 


4 


5 


6 


7 


8 





P 


.P. 


65 321 


33^ 


340 


350 


360 


369 


379 


389 


398 


408 






417 


427 


437 


446 


456 


466 


475 


485 


494 


504 


452 


514 


523 


533 


542 


552 


562 


571 


581 


590 


600 






453 


610 


619 


629 


^38 


648 


657 


667 


677 


686 


696 




10 


454 


705 


715 


724 


734 


744 


753 


763 


772 


782 


791 


.2 


2.0 


|455 


801 


810 


820 


830 


839 


849 


858 


868 


877 


887 


.3 


3-0 


456 


896 


906 


9^5 


925 


934 


944 


953 


963 


972 


982 






457 


991 


*OOI 


*oi3 


*020 


^029 


*o39 


*o48 


^058 


*o67 


*o77 


•4 

.5 


4.0 
5.0 


45 « 


66 085 


096 


105 


115 


124 


134 


143 


153 


162 


172 


.6 


6.0 


459 
460 

461 


181 


190 


200 


209 


219 


228 


238 


247 


257 


266 


•7 
.8 

•9 


7.0 
8.0 
9.0 


276 


285 


294 


304 


313 


323 


332 


342 


351 1 3^0 


370 


379 


389 


398 


408 


417 


426 


436 


445 


455 


462 


464 


473 


483 


492 


502 


511 


520 


530 


539 


548 






463 


558 


567 


577 


586 


595 


605 


614 


623 


^33 


642 






464 


652 


661 


673 


680 


689 


698 


708 


7x7 


726 


736 




a 


465 


745 


754 


764 


773 


782 


792 


801 


816 


820 


829 




9 


466 


838 


84S 


857 


S6e 


876 


885 


894 


904 


913 


922 


.1 

.2 


0.9 
1.9 

2.8 


467 


931 


941 


950 


959 


969 


978 


987 


996 


*oo6 


*oi5 


.3 


468 


67 024 


034 


043 


052 


061 


071 


080 


089 


099 


108 






469 
470 

471 


117 


126 


136 

228 


145 


154 


163 


173 


182 


191 


200 


• 4 

.5 
.6 

•7 
.8 


3.8 
4.7 
5.7 


210 

302 


219 


237 


246 


256 


265 


274 


283 


293 


311 


320 


329 


339 


348 


357 


3(^6 


376 


385 


472 


394 


403 


412 


422 


431 


440 


449 


458 


467 


477 


0.5 
7.6 


473 


486 


495 


504 


513 


523 


532 


541 


550 


559 


5^8 


•9 


8.5 


474 


578 


587 


596 


605 


614 


623 


^33 


642 


651 


660 






475 


669 


^78 


687 


697 


706 


715 


724 


733 


742 


75 J 






476 


760 


770 


779 


788 


797 


806 


815 


824 


^33 


842 






477 


852 


861 


870 


879 


8SS 


897 


906 


915 


924 


933 




9 


478 


943 


952 


961 


970 


979 


988 


997 


*oo6 


*oi5 


*024 


.1 

2 


0.9 

I 8 


479 

480 

481 


68 033 042 


051 


060 


070 
166 


079 


088 


097 


106 


115 


•3 

.4 

. 1; 


2.7 
3.6 

4. ^ 


124 


^33 


142 


151 


169 
259 


178 


187 


196 


205 


214 


223 


232 


241 


250 


268 


277 


286 


295 


482 


304 


3^3 


322 


331 


340 


349 


358 


367 


376 


385 


.6 


5.4 


483 


394 


403 


412 


421 


430 


439 


448 


457 


4^6 


475 






484 


484 493 


502 


511 


520 


529 


538 


547 


556 


565 


•7 

.8 


6.3 
7 2 


485 


574 583 


592 


601 


610 


61Q 


628 


637 


646 


654 


•9 


8.1 


480 


663 672 


681 


690 


699 


708 


717 


726 


735 


744 






487 


753 762 


770 


779 


788 


797 


806 


815 


824 


^33 






488 


842 


851 


860 


S6s 


877 


886 


895 


904 


913 


922 






489 
490 

491 


931 


940 


948 


957 


966 


975 


984 


993 


*002 


*oi5 


.1 

.2 

•3 


8 

o.S 
1-7 

2.5 


69 019 ! 02g 


037 


046 


055 


064 


073 


081 


090 


099 


108 


117 


126 


134 


143 


152 


161 


170 


179 


187 


492 


196 


205 


214 


223 


232 


240 


249 


258 


267 


276 






493 


284 


293 


302 


311 


320 


328 


337 


346 


355 


364 


.4 

.5 


3-4 

a. 2 


494 


372 


38^^ 


390 


399 


408 


41 6 


425 


434 


443 


451 


.6 


5-1 


495 


460 


469 


478 


487 


495 


504 


513 


522 


530 


539 






496 


548 


557 


565 


574 


583 


592 


600 


609 


618 


627 


.7 
8 


5.9 
6 A 


497 


635 


644 


653 


662 


670 


679 


688 


697 


705 


714 


•9 


7.6 


498 


723 


731 


740 


749 


758 


766 


775 


784 


792 


801 






499 
500 

N. 


810 


819 


827 


836 


845 


853 


862 


871 


879 


888 






897 


905 


914 


923 


931 


946 


949 


958 


966 


975 


1 


2 


3 


4 


5 6 


7 


8 


9 


P. 


P. 



332 









TABLE V.- 


-LOGARITHMS OF NUMBERS. 






500 





1 


»> 


•.i 


4 


»"> 


(> 


7 S 1) 


p 


P. 


69 897 
984 


905 


914 


923 


931 


946 

*027 


949 
*o36 


958 9^6 


975 
*o6i 






992 


*OOI 


*OIO 


*oi8 


*o44 *o53 


502 


70 070 


079 


087 


096 


105 


113 


122 


131 


139 


148 




9 

0.0 


5^3 


157 


165 


174 


182 


191 


200 


208 


217 


226 


234 


. I 


504 


243 


251 


260 


269 


277 


286 


294 


303 


312 


326 


.2 


1.8 


505 


329 


337 


346 


355 


3(^3 


372 


386 


389 


398 


406 


•3 


2.7 


506 


415 


423 


432 


441 


449 


458 


4^6 


475 


483 


492 


•4 
• 5 


3.6 
4-5 


507 


501 


509 


518 


526 


535 


543 


552 


566 


569 


578 


508 


586 


595 


603 


612 


626 


629 


637 


646 


654 


663 


.6 


5-4 


509 
510 

511 


672 


6Sd 


689 


697 


706 
791 


714 


723 


731 


740 


748 
833 
9J8 


•7 
.8 

•9 


6.3 

7-2 

S.I 


757 


765 


774 


782 


799 

884 


808 


816 


825 


842 


856 


859 


867 


876 


893 


901 


910 


512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*oo3 






5^3 


71 oil 


020 


028 


037 


045 


054 


062 


071 


079 


088 






514 


096 


105 


113 


121 


130 


138 


147 


155 


164 


172 






515 


186 


189 


197 


206 


214 


223 


231 


239 


248 


256 




8 


516 


265 


273 


282 


290 


298 


307 


315 


324 


332 


340 


. I 

.2 


0.8 
I -7 


517 


349 


357 


366 


374 


382 


391 


399 


408 


416 


424 


•3 


2.5 


5^^ 


433 


441 


449 


458 


465 


475 


483 


491 


500 


508 






519 
520 

521 


516 
600 


525 ' 


533 


542 


550 
633 


558 


567 


575 


583 


592 


•4 

•5 
.6 


3.4 
4.2 

5.1 


608 


617 


625 


642 


656 


659 667 


675 


684 


692 


700 


709 


717 


725 


734 


742 


750 


758 


522 


767 


775 


783 


792 


806 


808 


817 


825 


833 


842 


•7 

.8 


5-9 
6.8 


523 


850 


858 


867 


875 


883 


891 


900 


908 


916 


925 


•9 


7-6 


524 


933 


941 


949 


958 


9^6 


974 


983 


991 


999 


*oo7 




; 


525 


72 016 


024 


032 


040 


049 


057 


065 


074 


082 


090 




! 


526 


098 


107 


115 


123 


131 


140 


148 


156 


164 


173 






527 


181 


189 


197 


206 


214 


222 


230 


238 


247 


255 




8 


528 


263 


271 


280 


288 


296 


304 


312 


321 


329 


337 


.1 


o.S 
I 6 


529 
530 

r53i 


345 


354 


362 


370 


378 


386 


395 


403 


411 


419 


•3 
•4 


2.4 
3-2 

4..0 


427 


436 


444 


452 


466 


4^8 
55Q 


476 

558 


485 


493 


501 


509 


517 


526 


534 


542 


566 


575 


583 


532 


591 


599 


607 


615 


624 


632 


640 


648 


658 


664 


.6 


4.8 


533 


672 


68i 


689 


697 


705 


713 


721 


729 


738 


746 






534 


754 


762 


770 


778 


785 


795 


803 


811 


819 


827 


•7 
8 


5-6 
6.4 
7.2 


535 


835 


843 


851 


859 


868 


876 


884 


892 


900 


908 


■9 


53^ 


916 


924 


932 


941 


949 


957 


965 


973 


981 


989 






537 


997 


*oo5 


*oi3 


*02i 


*o3o 


*o3S 


*o46 


*o54 


*o62 


*o7o 






538 


73 078 


085 


094 


102 


1 10 


118 


126 


134 


143 


151 




' 


539 
540 

541 


159 
239 
3^9 


167 


175 


183 


191 


199 
279 


207 
287 
368 


215 


223 


231 


. I 

_ 2 

.3 


1 

0.7 

1-5 
2.2 


247 


255 


263 
344 


271 


295 


303 


311 


328 


336 


352 


360 


376 


384 


392 


542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 






543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


•4 


3-0 


544 


560 


568 


576 


584 


592 


600 


608 


615 


623 


631 


•5 
.0 


3-7 
4-5 


545 


639 


647 


655 


663 


671 


679 


687 


695 


703 


711 






546 


719 


727 


735 


743 


75^ 


759 


767 


775 


783 


791 


•7 
.8 

•9 


^.2 
6.0 
6.7 


547 


798 


806 


814 


822 


830 


838 


846 


854 


862 


870 


548 


878 


886 


894 


902 


909 


917 


925 933 


941 ! 949 






549 
550 

1 X. 


957 


965 


973 


981 


989 


997 

075 


*004 *OI2 


*026 


1*028 

107 






74 036 


044 


052 


060 


068 


083 091 1 099 


. 


1 


! 2 


1 3 


1 4 


5 


C, 7 S 


P 


. P. 



333 









TABLE V.- 


-LOGARITHMS OF NUMBERS. 






650 

551 


1 1 


2 


3 


4 


5 


6 


7 


8 


9 


P^ 


P. 


74 036 
n5 


044 


052 


060 


068 


075 


083 


091 


099 

178 


107 
186 






123 


131 


139 


146 


154 


1O2 


170 


55- 


194 


202 


209 


217 


225 


233 


241 


249 


257 


264 






553 


272 


280 


288 


296 


304 


312 


319 


327 


335 


343 






554 


351 


359 


S'^G 


374 


3S2 


390 


398 


406 


413 


421 






55^ 
5S(^ 


429 


437 


445 


453 


466 


463 


476 


484 


492 


499 






507 


515 


523 


531 


538 


546 


554 


562 


570 


577 




8 


557 


585 


593 


601 


609 


616 


624 


62,2 


640 


648 


655 


.1 


0.8 

T 6 


55S 


663 


671 


679 


687 


694 


702 


710 


718 


725 


733 


.3 


2.4 


559 


741 
819 

896 


749 


756 


764 


772 
850 


780 


788 


795 


803 


811 


•4 
•5 
.6 


3-2 
4.0 
4.8 


560 


826 


834 


842 


857 


865 


873 


881 


888 


5^1 


904 


912 


919 


927 


935 


942 


950 


958 


966 


562 


973 


981 


989 


997 


'•004 


*OI2 


*020 


*027 


*o3,S 


*o43 






5^3 


75 051 


058 


065 


074 


081 


089 


097 


105 


112 


120 


•7 


5-6 

fi A 


565 
566 


128 


135 


143 


151 


158 


166 


174 


182 


189 


197 


•9 


D.4 
7.2 


205 


212 


220 


228 


235 


243 


251 


258 


266 


274 






281 


289 


297 


304 


312 


320 


327 


335 


343 


350 






5^7 
568 

569 
670 

571 


358 


366 


*T -7 

J/3 


38i 


389 


396 


404 


412 


419 


427 






435 


442 


450 


458 


465 


473 


4S0 


488 


496 


503 






511 


5^9 


525 


534 


541 


549 


557 


564 


572 


580 




1 

^ 


587 


595 


602 


6 10 


618 


625 


633 


641 


648 656 


663 


671 


679 


6-6 


694 


701 


709 


717 


724 


732 


!57-^ 


739 


747 


755 


762 


770 


M ^ •? 

ill 


785 


7Q2 


800 


808 


. I 


0.7 

1-5 

2.2 


573 


815 


823 


830 


838 


846 


853 


861 


868 


876 


883 


• 3 


574 


891 


899 


906 


914 


921 


929 


936 


944 


9Si 


959 






575 


967 


974 


982 


989 


997 


^■"004 


*OI2 


^019 


*027 


*o34 


•4 


3-9 


570 


76 042 


050 


057 


065 


072 


oSo 


087 


095 


102 


no 


•5 
.6 


3-7 
4.5 


577 


117 


125 


132 


140 


147 


155 


162 


170 


178 


185 






57S 


193 


200 


208 


2^5 


223 


230 


238 


245 


253 


260 


•7 

.8 

•9 


5-2 

6.0 
6.7 


579 
680 

581 


268 


275 


283 


290 


298 


305 
380 


313 

387 


320 


328 


335 


343 


350 


358 


365 


372 


395 

470 


4c 2 


410 


417 


425 


432 


440 


447 


455 


462 


477 


485 


582 


492 


500 


507 


514 


522 


529 


537 


544 


552 


559 






5^3 


567 


574 


582 


589 


596 


604 


6ii 


619 


626 


634 






^5H 


641 


648 


656 


663 


671 


678 


686 


693 


700 


708 






|5^5 


715 


723 


730 


738 


745 


752 


760 


767 


775 


782 






5S0 


790 


797 


804 


812 


819 


827 


534 


841 


849 


856 




7 


i5«7 


864 


871 


878 


SS6 


893 


901 


908 


9LS 


923 


930 


. I 


0.7 


588 
589 
590 

591 


937 


945 


952 


960 


967 


974 


982 


989 


997 


'^'004 


.3 


1.4 
2.1 


77 on 


019 


026 


^33 


041 
114 


048 


055 
129 


063 


070 


078 

151 

225 


.4 
•5 
.6 


2.8 

3-5 
4.2 


085 
158 


092 


100 


107 


122 
195 


136 
210 


144 


166 


173 


181 


188 


2C3 


217 


592 


232 


239 


247 


254 


261 


269 


276 


283 


2QI 


298 






593 


305 


313 


320 


327 


335 


342 


349 


3S6 


364 


--* 

371 


•7 

s 


4-9 

^ 6 


594 


378 


386 


393 


400 


408 


415 


422 


430 


437 


444 


.9 


6.3 


|595 


451 


459 


465 


473 


481 


488 


495 


S^3 


510 


517 






59t> 


524 


532 


539 


546 


554 


561 


S^d> 


575 


583 


590 






1597 


597 


604 


612 


619 


626 


634 


641 


648 


6SS 


667, 






!598 


670 


677 


684 


692 


699 


706 


713 


721 


728 


735 






j599 
600 

N. 


742 


750 


757 


764 


771 


779 


786 


793 


8o5 


808 






815 


822 


829 


837 


844 


851 


858 


S66 


873 


880 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P, 


p- 



334 









TABLE V.- 


-LOGARITHMS ( 


3F NUMBERS. 






N. 





1 




.ad 


1 3 1 4 


n a 7 


S 1) 


P 


. P. 


600 

6oi 
602 
603 


77 ^15 
887 

959 

78 031 


822 


829 


1 ^37 


844 


85 i 


858 


866 


873 


8S0 






894 

1 967 

039 


902 

974 
046 


909 
981 

053 


9^6 

988 
060 


923 

995 
067 


931 

*oo3 

075 


938 

*OIO 

082 


945 
*oi7 

089 


952 

*024 

096 


604 
605 

606 


103 

175 
247 


1 1 1 

, 182 

254 


118 
190 
261 


125 
197 
269 


132 
204 
276 


139 
211 

283 


147 

218 

290 


154 
226 
297 


161 

233 

304 


168 

240 

311 




1 


607 
608 
609 


319 

390 
461 


i 3^^ 

\ 397 
469 


333 
404 

476 


340 
412 

483 


347 
419 

490 


354 
426 

568 


362 

433 

504 


369 

446 

511 


376 
447 
518 


383 

454 
526 


.1 
.2 

•3 

•4 
.5 
.6 

■7 

8 


0.7 

1-5 
2.2 

3.0 
3-7 

4.5 

5.2 
6 


610 


533 
604 

675 
746 


540 


1 547 


554 


J6i 
632 

703 
774 


575 


583 


590 


597 


611 
612 
613 


, 61 1 
682 

753 


618 
689 
760 


625 

696 

767 


639 

716 

781 


646 
717 

788 


654 

725 
795 


661 

732 
802 


668 

739 
8jo 


614 

615 
616 


817 

887 
958 


824 
894 

965 


831 

901 
972 


838 

908 
979 


845 
915 
986 


852 
923 
993 


859 

930 

*ooo 


866 

937 

*oo7 


873 

944 

*oi4 


886 
951 

*02I 


.9 


6.7 


617 
618 
619 


79 028 
099 
169 


035 
106 

176 


042 

113 

183 


049 
120 
190 

260 


056 
127 

197 

267 


063 

134 
204 


076 

141 

211 


078 
148 
218 


085 

155 
225 


092 
162 
232 


.1 

.2 

•3 


7 

0.7 
1.4 
2.1 


620 


239 

309 
379 
449 


246 


253 


274 


281 


288 


295 


302 


621 
622 
623 


316 
386 
456 


323 

393 
462 


330 

400 
469 


337 

407 

476 


344 
414 

483 


351 
421 

490 


358 
428 

497 


365 
435 

504 


372 
442 
511 


624 
625 
626 


518 

588 

657 


595 
664 


532 
602 
671 


539 
609 

678 


546 
6r6 

685 


553 
622 

692 


560 
629 
699 


567 

^36 

706 


574 
643 
713 


581 
656 

720 


•4 
•5 
.6 


2.8 

3-5 
4.2 


627 
628 
629 


727 
796 
865 


733 

803 

872 


740 
810 
879 


74? 
8i§ 

S86 


754 
823 

892 


761 

830 

_89i_ 

968 


768 
837 
9O6 


775 
844 
913 


782 

85 f 
920 


789 
858 
927 


•7 
.8 

•9 


4.9 
5-6 
6.3 


680 


934 


941 ! 


948 


954 


961 


975 


982 


989 
058 

126 

195 


996 
065 

^33 

202 


631 
632 

^33 


80 003 
071 
140 


010 

078 
147 


016 
085 
154 


023 
092 
i6i 


036 
099 
168 


037 
106 

174 


044 

113 
iSi 


051 

120 

188 


634 
635 
636 


209 

277 

345 


216 
284 
352 


222 
291 

359 


229 
298 
366 


236 
304 
373 


243 
311 
380 


250 

3^8 
386 


257 
325 
393 


263 

332 
406 


270 

339 

407 




6 


637 
639 


414 

482 

550 

618 

686 

753 
821 


421 
489 

557 


427 

495 
563 


434 
502 

570 


441 

509 

577 


448 
516 
584 


455 
523 
591 


461 

529 
597 


468 

536 
604 

672 


475 

543 
611 

679 

746 

814 

882 


.1 

.2 
•3 

.4 
•5 
.6 

•7 

8 


0.6 

1-3 
1.9 

.6 

3-2 

3-9 

4-5 
5-2 

5-S 


640 


625 


631 


638 


645 


652 


658 


665 


641 
642 

643 


692 
760 
S2S 


699 

767 
834 


706 

774 
841 


713 

780 

848 


719 

787 

855 


726 
794 
86: 


733 
801 

868 


740 
807 

875 


644 

645 
646 


888 

956 
81 023 


895 
962 ; 

030 


902 
969 

^36 


909 
976 

043 


915 

983 

050 


922 
989 

057 


929 

996 
063 


936 

*oo3 

076 


942 

*OIO 

077 


949 
*oi6 

083 


•9 


647 
648 
649 


090 

157 
224 


097 
164 
231 


104 
171 
238 


no 

' 177 
244 


117 

184 

251 


124 
191 

258 


135 
197 
264 


137 
204 
27! 


144 
211 

278 

345 


151 
218 

284 

351 






650 


291 


298 


304 


311 


3^8 


324 33^ 


338 


N. 





1 1 


2 


3 


4 


5 1 


7 8 i 9 1 


P. 


P. 



335 



TABLE v.— LOGARITHMS OF NUMBERS. 





N. 





1 


2 


3 


4 


5 


6 


7 1 8 


9 


P.P. 




650 

|65i 


81 291 


298 


304 


311 


318 


324 


33^ 


33^ 


345 


351 


' 




358 


365 


371 


378 


3^5 


391 


398 


405 


411 


418 




652 


425 


431 


438 


444 


451 


458 


464 


471 


478 


484 






(>5S 


491 


498 


504 


511 


518 


524 


531 


538 


544 


551 






654 


558 


564 


571 


577 


584 


591 


597 


604 


611 


617 






655 


624 


631 


637 


644 


650 


657 


664 


676 


677 


684 


j 




656 


696 


697 


703 


710 


717 


723 


730 


736 


743 


750 




7 




657 


756 


763 


770 


776 


783 


789 


796 


803 


809 


816 


.1 


0.7 1 




658 


822 


829 


836 


842 


849 


855 


862 


869 


875 


882 




1.4 
2. 1 




659 
660 

661 


888 


895 


901 


908 


915 


921 


928 


934 


941 


948 


•4 

•5 
.6 


2.8 

3.5 
4.2 




954 


961 


967 


974 


986 


987 


994 


*oo6 


*oo7 


*oi3 




82 020 


^^6. 


<^33 


040 


046 


053 


059 


066 


072 


079 




662 


086 


092 


099 


105 


112 


118 


125 


131 


^3^ 


145 








663 


151 


158 


164 


171 


177 


184 


190 


197 


203 


210 


•7 

« 


4.9 




664 


217 


223 


230 


236 


243 


249 


256 


262 


269 


275 


•9 


5'0 
6.3 




665 


282 


288 


295 


302 


308 


3^5 


321 


328 


334 


341 






666 


347 


354 


360 


367 


373 


380 


386 


393 


399 


406 






667 


412 


419 


425 


432 


438 


445 


451 


458 


464 


471 






668 


477 


484 


490 


497 


5^3 


510 


516 


523 


529 


53<^ 






1 669 
670 

,671 


542 


549 


555 


562 


568 


575 


581 


588 


594 


601 


9 




607 


614 


620 


627 


^33 


640 


646 


653 


659 


666 




672 


678 


685 


691 


698 


704 


711 


717 


724 


730 






A 2 




'672 


737 


743 


750 


756 


763 


769 


775 


782 


788 


795 


.2 


0-6 
1 .3 




673 


801 


808 


814 


821 


827 


834 


840 


846 


853 


859 


•3 


1.9 




'674 


866 


872 


879 


885 


892 


898 


904 


911 


917 


924 




2.6 

3-2 

3.9 




675 


930 


937 


943 


949 


956 


962 


969 


975 


982 


988 


•4 




676 


994 


*OOI 


*oo7 


*oi4 


*020 


*027 


*o33 


*o39 


*046 


*052 


.6 




677 


83 059 


065 


071 


078 


084 


091 


097 


103 


no 


116 








678 


123 


129 


136 


142 


148 


155 


161 


168 


174 


180 


•7 

s 


4-5 

5-2 

5-8 




679 
680 

j68i 


187 
251 


193 


200 


206 


212 


219 


225 


231 


238 


244 


•9 




257 


263 


270 


276 


283 


289 


295 


302 


308 






314 


321 


327 


334 


340 


346 


353 


359 


365 


372 




'682 


378 


385 


391 


397 


404 


410 


416 


4.23 


429 


435 






I683 


442 


448 


455 


461 


467 


474 


480 


486 


493 


499 






684 


50^ 


512 


518 


524 


531 


537 


543 


550 


556 


562 






685 


569 


575 


581 


S88 


594 


6o5 


607 


613 


619 


626 


A 




6S6 


632 


^38 


645 


651 


657 


664 


676 


676 


683 


689 


J 


6 




i687 


695 


702 


708 


714 


721 


727 


733 


740 


746 


752 


.2 


1.2 




68S 


759 


765 


771 


778 


784 


790 


796 


803 


809 


815 


•3 


1.8 




689 
690 

691 


822 
885 


828 


834 


841 


847 


853 


859 


866 


872 


878 


.4 

• 5 
.6 


2.4 

3-0 
3.6 




891 


897 


904 


910 


916 


922 


929 


935 


941 




948 


954 


960 


966 


973 


979 


985 


992 


998 


*oo4 




692 


84 010 


017 


023 


029 


035 


042 


048 


054 


061 


067 


•7 

.8 


4.2 
4.8 




693 


073 


079 


086 


092 


098 


104 


III 


117 


123 


129 




694 


136 


142 


148 


154 


161 


167 




179 


186 


192 


•9 


5.4 




695 


198 


204 


211 


217 


223 


229 


236 


242 


248 


254 






696 


261 


267 


273 


279 


286 


292 


298 


304 


311 


317 






697 


323 


329 


335 


342 


348 


354 


360 


367 


373 


379 






698 


385 


392 


398 


404 


410 


416 


423 


429 


435 


441 






699 
700 


447 

510 


454 


460 


465 


472 


479 


485 


491 

553 


497 


5^3 






516 


522 


528 


534 


541 


547 


559 i 


565 


L 


N. 





1 


2 


3 


4 


5 


6 7 


8 9 P. P, 1 



336 









TABLE V.- 


-LOGARITHMS OF NUMBERS. 






700 

701 





1 ! 12 


! .'i " 


4 


5 


(; 


7 


s 


*) 


!• 


. P. 


84 510 
572 


516 


522 


528 


534 


541 


547 


553 


559 


565 






578 


584 


590 


596 


603 


609 


615 


621 


627 


702 


(>?>l 


640 


646 


652 


658 


664 


671 


677 


683 


689 






,703 


695 


701 


708 


714 


720 


726 


732 


739 


745 


751 






704 


757 


763 


769 


776 


782 


788 


794 


806 


806 


813 






705 


819 


825 


831 


837 


843 


849 


856 


862 


868 


874 






706 


880 


886 


893 


899 


905 


911 


917 


923 


929 


936 




6 


707 


942 


948 


954 


960 


9^6 


972 


979 


985 


991 


997 


. I 
.2 


0.6 
1 .3 


708 


85 003 


009 


015 


021 


028 


034 


040 


046 


052 


05 8 


•3 


1.(3 


1709 
710 

711 


064 


070 


077 


083 


089 


095 

156 
217 


lOI 

162 


107 
163 


113 


119 


•4 
•5 
.6 


2.6 

3-2 

3.9 


126 


132 


138 


144 


150 


174 


181 


187 


193 


199 


205 


211 


223 


229 


236 


242 


712 


248 


254 


260 


265 


272 


278 


284 


290 


297 


303 




^ 


713 


309 


315 


321 


327 


ZZ2> 


339 


345 


351 


357 


Z^?> 


•7 

.8 


4-5 

5-2 1 


714 


370 


376 


382 


388 


394 


400 


406 


412 


418 


424 


•9 


5-8 


715 


430 


436 


443 


449 


455 


461 


467 


473 


479 


485 






716 


491 


497 


503 


509 


515 


521 


527 


533 


540 


546 






717 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 






718 


612 


618 


624 


635 


^36 


642 


648 


655 


661 


667 






719 
720 

721 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 




6 

n f\ 


733 


739 


745 


751 


757 


763 


769 


775 


781 


787 


793 


799 


805 


8ii 


817 


823 


829 


835 


841 


847 


722 


853 


859 


865 


872 


878 


884 


890 


896 


902 


908 


.2 


I .2 


1723 


914 


920 


926 


932 


938 


944 


95Q 


956 


962 


968 


•3 


I. 8 


724 


974 


980 


986 


992 


998 


*oo4 


*OTO 


*oi6 


*022 


*028 






i725 


86034 


040 


046 


052 


058 


063 


069 


075 


081 


087 


•4 


2.4 
3.0 


1726 


093 


099 


105 


III 


117 


123 


129 


135 


141 


147 


.6 


3.6 


J727 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 






728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


•7 
.8 


4.2 
4.8 1 


729 
730 

731 


273 
332 
391 


278 


284 


290 


296 


302 


3O8 


314 


320 


326 


'9 


5.4 


338 
397 


344 


350 


356 


362 


368 


374 


380 


386 


403 


409 


415 


421 


427 


433 


439 


445 


732 


451 


457 


463 


469 


475 


481 


486 


492 


498 


504 






733 


510 


S'^G 


522 


528 


534 


540 


546 


552 


558 


563 






|734 


569 


575 


58I 


587 


593 


599 


605 


611 


617 


623 






735 


623 


634 


640 


646 


652 


658 


664 


670 


676 


682 




5 


736 


688 


693 


699 


705 


711 


717 


723 


729 


735 


741 




737 


746 


752 


758 


764 


770 


776 


782 


788 


794 


800 


. I 
.2 


0.5 
I . I 


738 


805 


8ii 


817 


823 


829 


835 


841 


847 


852 


858 


•3 


1-6 


,739 
740 

741 


864 

923 
982 


870 

929 
987 


876 


882 


888 


894 


899 


905 


911 


917 
976 

*o34 


.4 

• 5 
.6 


2.2 

2.7 
3-3 


935 


941 


946 


952 

*OII 


958 


964 


970 


993 


999 


*oo5 


*oi7 


*023 


*028 


742 


87 040 


046 


052 


058 


064 


069 


075 


o8i 


087 


093 




^ 


743 


099 


104 


no 


116 


122 


128 


134 


140 


145 


151 


•7 
.8 


3-8 
4.4 


744 


157 


163 


169 


175 


180 


1 86 


192 


198 


204 


210 


•9 


4.9 


745 


215 


221 


227 


233 


239 


245 


250 


256 


262 


268 






'746 


274 


279 


285 


291 


297 


Z^3 


309 


314 


320. 


326 






747 


332 


338 


343 


349 


355 


361 


367 


372 


378 


384 






748 


390 


396 


402 


407 


413 


419 


425 


431 


436 


442 






749 
750 


448 


454 


460 


465 


471 
529 


477 


483 


489 
546 


494 


500 
5:s8 




: 


506 


512 


517 


523 


535 


541 


552 


" 





1 2 


3 


4 


5 


G 


7 


8 


J) 


P. 


P. 



337 









TABLE V.- 


-LOGARITHMS OF NUMBERS. 






N. 


1 1 


2 


3 1 4 


5 1 


6 


7 


8 


9 


P. 


P. 


750 

1751 


87 ^06 ! 512 


517 


523 


529 


535 


541 


546 


552 


558 




1 


564 


570 


575 


58i 


587 


593 


598 


604 


610 


616 


'752 


622 


627 


^33 


639 


645 


650 


656 


662 


668 


673 






;753 


679 


685 


691 


697 


702 


708 


714 


720 


725 


731 






!754 


737 


743 


748 


754 


760 


766 


771 


777 


783 


789 






755 


794 


800 


806 


812 


817 


823 


829 


835 


840 


846 






75t» 


852 


858 


863 


869 


875 


881 


886 


892 


898 


904 




6 


757 


909 915 


921 


927 


932 


938 


944 


949 


955 


961 


.1 
2 


0.6 
I 2 


I75S 


967 


972 


978 


984 


990 


995 


*ooi 


"^007 


*OI2 


*oi8 


.3 


i.'s 


1759 
i760 

761 


88 024 


030 


035 


041 


047 
1 04 


053 


058 


064 


070 


075 
133 


•4 
.5 
.6 


2.4 

3.0 
3.6 


081 


087 


093 


098 


no 


115 


121 

178 


127 


138 


144 


150 


155 


161 


167 


172 


184 


190 


762 


195 


201 


207 


212 


218 


224 


229 


235 


241 


247 






\7^3 


252 


258 


264 


269 


275 


281 


286 


292 


298 


303 


■7 
« 


4.2 

4.8 
5-4 


1764 


309 


315 


320 


3^6 


332 


337 


343 


349 


355 


366 


•9 


i7t^5 


366 


372 


377 


383 


389 


394 


400 


406 


411 


417 






766 


423 


428 


434 


440 


445 


451 


457 


462 


468 


474 






7^7 


479 


485 


491 


496 


502 


508 


513 


519 


525 


530 






I708 


536 


542 


547 


553 


558 


564 


570 


575 


58i 


587 






(709 
770 

1771 


592 


598 


604 


609 


615 
671 


621 


626 


632 


638 


643 




s 


649 


654 


660 


666 


677 


683 


688 


694 


700 


705 


711 


716 


722 


728 


733 


739 


745 


750 


756 


i772 


761 


767 


773 


778 


784 


790 


795 


801 


806 


812 


.1 


0-5 


773 


818 


823 


829 


835 


846 


846 


851 


857 


863 


S6s 


.3 


1 . I 

1-6 


774 


874 


879 


885 


891 


896 


902 


907 


913 


919 


924 






!775 


930 


936 


941 


947 


952 


958 


964 


969 


975 


980 


•4 


2,2 


|77^ 


986 


992 


997 


*oo3 


*oo8 


*oi4 


*oi9 


*025 


*o3i 


*o36 


•5 
.6 


2.7 
3-3 


|777 


89 042 


047 


053 


059 


064 


070 


075 


081 


087 


092 






'778 


098 


103 


109 


114 


120 


126 


131 


137 


142 


148 


.7 


3-8 


779 
1780 

1781 


153 


159 


165 


170 


176 


181 


187 


193 

248 


198 


204 


. 
•9 


4-4 i 
4.9 


209 215 


226 


226 


231 


237 


243 


254 


259 


265 


276 


276 


282 


287 


293 


298 


304 


309 


315 


1782 


320 


326 


332 


337 


343 


348 


354 


359 


365 


370 






|7«3 


376 


38i 


387 


393 


398 


404 


409 


415 


420 


426 






1784 


431 


437 


4J2 


448 


454 


459 


465 


470 


476 


481 






^785 


487 


492 


498 


503 


509 


514 


520 


525 


531 


536 






786 


542 


548 


553 


559 


564 


570 


575 


581 


586 


592 




5 


787 


597 


603 


60s 


614 


619 


625 


630 


636 


641 


647 


.1 


0.5 


788 


652 


658 


66s 


669 


674 


680 


685 


691 


696 


702 


.3 


1-5 


789 
790 

791 


707 


7^3 


718 


724 


729 


735 


740 


746 


751 


757 


•4 
•5 
.6 


1 
2.0 

2.5 ' 
3-0 ! 


762 


768 


773 


779 


784 


790 


795 


801 


806 


812 
867 


817 


823 


828 


834 


839 


845 


856 


856 


861 


792 


872 


878 


883 


889 


894 


900 


905 


911 


916 


922 






793 


927 


933 


938 


943 


949 


954 


960 


965 


971 


976 


.7 
.8 

•9 


3-5 


794 


982 


987 


993 


998 


*oo4 


*oo9 


*o,5 


*020 


*026 


*o3i 


4.0 
4-5 


795 


90 036 


042 


047 


053 


058 


064 


069 


075 


080 


086 






796 


091 


097 


102 


107 


113 


118 


124 


129 


135 


140 






797 


146 


151 


156 


162 


167 


173 


178 


184 


189 


195 






798 


200 


205 


211 


216 


222 


227 


233 


238 


244 


249 






799 
800 


254 


260 


265 


271 


276 


282 


287 


292 


298 


3^3 






309 


314 


320 


325 


330 


33^ 


341 


347 


352 


358 


1 N. 





1 


2 


3 1 4 


5 


6 


7 


8 


9 


P 


.P. 

1 



338 









TABLE V.- 


-LOGARITHMS OF NUMBERS. 






1800 

8oi 





1 


2 


3 


4 


5 <; 7 


S 


9 


P. P. 


90 309 


314 


320 


325 


330 


336 


341 


347 


352 


358 




363 


3^8 


374 


379 


385 


390 


396 


401 


406 


412 


802 


417 


423 


428 


433 


439 


444 


450 


455 


466 


466 




803 


471 


477 


482 


488 


493 


498 


504 


509 


515 


520 




804 


525 


531 


536 


542 


547 


552 


558 


563 


569 


574 




805 


579 


585 


590 


596 


601 


6og 


612 


617 


622 


628 




;8o6 


633 


639 


644 


649 


655 


666 


666 


671 


^76 


682 




807 


687 


692 


698 


703 


709 


714 


719 


725 


730 


736 




808 


741 


746 


752 


757 


762 


768 


773 


778 


784 


789 




809 
810 

811 


795 


800 


805 


811 


816 


821 


827 


832 


838 
891 


843 


a 


848 


854 


859 


864 


870 


875 


886 


886 


896 


902 


907 


913 


918 


923 


929 


934 


939 


945 


950 




b 


812 


955 


961 


9^6 


971 


977 


982 


987 


993 


998 


*oo3 


, I 
.2 


0. 3 
I.I 


«i3 


91 009 


014 


019 


025 


030 


036 


041 


046 


052 


057 


•3 


1-6 


814 


062 


068 


073 


078 


084 


089 


094 


100 


105 


1 16 






815 


116 


121 


126 


131 


137 


142 


147 


153 


158 


163 


•4 
.6 


2 . 2 

2. 7 


816 


169 


174 


179 


185 


190 


195 


201 


206 


211 


217 


3-3 


817 


222 


227 


233 


238 


243 


249 


254 


259 


264 


270 






818 


275 


280 


286 


291 


296 


302 


307 


312 


318 


323 


•7 

8 


3-8 
4-4 

4.Q 


819 
820 

821 


328 
381 


333 


339 


344 


349 

402 


355 


360 


365 
418 
471 


371 


376 


•9 


3H 


392 


397 


408 


413 


423 


429 




434 


439 


445 


450 


455 


461 


466 


476 


482 


1822 


487 


492 


497 


503 


508 


513 


519 


524 


529 


534 




823 


540 


545 


550 


556 


561 


5^6 


571 


577 


582 


587 




824 


592 


598 


603 


608 


614 


619 


624 


629 


635 


640 




1825 


645 


655 


656 


661 


666 


671 


677 


682 


687 


692 




826 


698 


703 


708 


714 


719 


724 


729 


735 


740 


745 




827 


750 


756 


761 


766 


771 


777 


782 


787 


792 


798 




828 


803 


808 


813 


819 


824 


829 


834 


839 


84s 


8so 




829 
830 

831 


855 
908 

960 


860 


S66 


871 


876 


881 


887 


892 


897 


902 


f 


913 


918 


923 


928 


934 


939 


944 


949 


955 


965 


976 


976 


981 


986 


991 


996 


*002 


*oo7 


I 




0.5 
I.O 


832 


92 012 


017 


023 


028 


033 


038 


043 


049 


054 


059 


.2 


833 


064 


069 


075 


080 


085 


090 


096 


lOI 


106 


III 


.3 


1-5 


834 


116 


122 


127 


132 


137 


142 


148 


153 


158 


163 






835 


168 


174 


179 


184 


189 


194 


200 


205 


210 


215 


•4 

.5 


2.5 


836 


220 


226 


231 


236 


241 


246 


252 


257 


262 


267 


.6 


30 


837 


272 


277 


283 


288 


293 


298 


3^3 


309 


314 


319 






838 


324 


329 


335 


340 


345 


350 


355 


366 


366 


37^ 


•7 
.8 


3-5 
4.0 


839 
i840 

1841 


376 


381 


386 


391 


397 


402 


407 


412 


417 


423 
474 


•9 


4.5 


428 


433 


438 
490 


443 


448 
500 


454 


459 


464 
515 


469 


i 


479 


485 


495 


505 


510 


521 


526 


I842 


531 


536 


541 


546 


552 


557 


562 


567 


572 


577 




843 


583 


588 


593 


598 


603 


608 


613 


619 


624 


629 




844 


634 


639 


644 


649 


655 


660 


665 


670 


675 


686 




1845 


685 


691 


696 


701 


706 


711 


716 


721 


727 


732 




846 


737 


742 


747 


752 


757 


762 


768 


773 


778 


783 




'847 


788 


793 


798 


803 


809 


814 


819 


824 


829 


834 




1848 


839 


844 


850 


855 


860 


865 


876 


875 


886 


885 




'849 
850 


891 
942 


896 
947 


901 


906 


911 


9^6 


921 


926 

977 


931 

982 


937 

988 




952 


957 


962 


967 


972 


1 ^• 





1 


2 


3 4 


5 


G 


7 


8 





P. P. 



339 











TABLE V.- 


-LOGARITHMS OF NUMBERS. 








850 

851 





1 


2 


3 


4 5 1 6 1 7 i 


8 9 1 


P. 


P. 1 




92 942 


947 


952 


957 


962 


967 


972 


977 


982 


988 








993 


998 


*oo3 


*oo8 


*oi3 


*oi8 


*023 


*028 


*o34 


*o39 




852 


93 044 


049 


054 


059 


064 


069 


074 


079 


084 


090 








853 


095 


100 


105 


no 


115 


120 


125 


130 


135 


140 








854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


191 








855 


196 


201 


207 


212 


217 


222 


227 


232 


237 


242 






1 


856 


247 


252 


257 


262 


267 


272 


278 


283 


288 


293 




s 




857 


298 


303 


308 


313 


318 


323 


328 


333 


338 


343 


.1 

2 


0.5 

I I 




858 


348 


354 


359 


364 


369 


374 


379 


384 


389 


394 


.3 


1.6 




859 
860 

861 


399 


404 


409 


414 


419 


424 


429 


434 


439 


445 


•4 
•5 
.6 


2.2 

2.7 
3.3 




450 


455 


460 


465 


470 


475 


480 


485 


490 


495 




506 


505 


510 


515 


526 


525 


530 


535 


540 


545 




862 


550 


556 


561 


566 


571 


576 


581 


586 


591 


596 








863 


601 


606 


611 


616 


621 


626 


631 


636 


641 


646 


•7 


3-8 




864 


. 65J 


656 


661 


66s 


671 


676 


681 


686 


691 


696 


•9 


4.4 
4.9 




865 


701 


706 


711 


716 


721 


726 


731 


736 


742 


747 








866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 








867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 








868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 








869 

870 
871 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




5 




952 
94 002 


957 962 1 967 


972 


977 


982 


987 


992 


997 




007 012 


017 


022 


025 


031 


^36 


041 


046 




872 


051 


056 


061 


065 


071 


076 


081 


086 


091 


096 


.1 
2 


0.5 
I 




873 


lOI 


105 


III 


116 


121 


126 


131 


136 


141 


146 


.3 


1.5 




874 


151 


156 


161 


166 


171 


176 


181 


186 


191 


196 








875 


201 


206 


210 


215 


226 


225 


236 


235 


246 


245 


.4 


2.0 




876 


250 


255 


260 


265 


270 


275 


280 


285 


290 


295 


•5 
.6 


2.5 
3.0 




877 


300 


305 


310 


315 


320 


324 


329 


334 


339 


344 








878 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


• 7 


3-5 




879 

880 

881 


399 


404 


409 


413 


418 
468 


423 


428 


433 


438 


443 


.0 
•9 


4.0 
4.5 




448 ' 4S3 


458 i 463 


473 


478 


483 


487 


492 




497 


502 


507 


512 


517 


522 


527 


532 


537 


542 




882 


547 


552 


556 


56i 


566 


571 


576 


58i 


586 


591 








I883 


596 


601 


606 


611 


615 


626 


625 


636 


635 


646 




' 


|884 


645 


650 


655 


660 


665 


670 


674 


679 


684 


689 








885 


694 


699 


704 


709 


714 


719 


724 


728 


733 


738 








886 


743 


748 


753 


758 


763 


768 


773 


777 


782 


787 




4 




887 


792 


797 


802 


807 


812 


817 


821 


826 


83I 


836 


.1 


0.4 




888 


841 


846 


851 


856 


861 


865 


870 


875 


886 


885 


.3 


U.9 
1-3 




889 
890 
891 


890 


895 


900 


905 


909 


914 


919 


924 


929 


934 


.4 
.6 


1.8 
2.2 
2.7 




939 


944 


949 


1 953 


958 


963 


968 


973 


978 


983 




988 


992 


997 j*002 


*oo7 


*OI2 


*oi7 


*022 


*026 


031 




892 


95 036 


041 


046 


051 


056 


061 


065 


070 


075 


o85 








893 


085 


090 


095 


099 


104 


109 


114 


119 


124 


129 


.7 
g 


3-1 
3.6 
4.6 




894 


134 


138 


143 


148 


153 


158 


163 


167 


172 


177 


•9 




895 


182 


187 


192 


197 


201 


206 


211 


216 


221 


226 








896 


231 


235 


240 


245 


250 


255 


260 


264 


269 


274 








897 


279 


284 


289 


294 


298 


3<=>3 


308 


3^3 


318 


323 








898 


327 


332 


337 


342 


347 


352 ! 356 


361 


366 


371 








899 
900 


376 


381 


385 


390 


395 


400 ! 405 


410 


414 


419 








424 


429 


434 


438 


443 


448 1 453 


458 


463 


467 







1 


2 


3 


4 


5 ! 6 


7 8 


9 


P 


, P. 



340 



TABLE v.— LOGARITHMS OF NUMHERS. 



900 

901 





1 


12 


3 


4 


»"> 


(i 


7 


S 





l\ l». . 


95 424 

472 


429 


434 


438 


443 
492 


448 


453 


458 


4^J3 


467 
516 




477 


482 


487 


496 


501 


506 


511 


902 


520 


525 


530 


535 


540 


544 


549 


554 


559 


564 




903 


569 


573 


578 


5 ^^3 


588 


593 


597 


602 


607 


612 




904 


617 


621 


626 


63? 


636 


641 


^M5 


650 


655 


660 




905 


665 


669 


674 


679 


684 


689 


^93 


^98 


703 


708 




906 


713 


717 


722 


727 


732 


737 


741 


746 


751 


756 




907 


760 


765 


770 


775 


780 


784 


789 


794 


799 


804 




908 


808 


813 


8i8 


823 


827 


832 


837 


842 


847 


851 




909 
910 
911 


^56 


861 


866 


870 


875 
923 


880 


885 


890 


894 
942 


899 




904 
952 


_9°9_ 
956 


9^3 


918 


928 


933 

986 


937 


947 


961 


966 


971 


975 


985 


990 


994 




5 1 


912 


999 


*oo4 


"^009 


*oi4 


*oi8 


^023 


*028 


*^33 


*o37 


*042 


.1 


0.5 


913 


96 047 


052 


056 


061 


066 


071 


075 


086 


085 


090 


.3 


1-5 


914 


094 


099 


104 


109 


113 


1^8 


123 


128 


132 


137 






915 


142 


147 


151 


156 


161 


166 


170 


175 


180 


185 


•4 


2.0 


916 


189 


194 


199 


204 


208 


213 


218 


222 


227 


232 


•5 
.6 


2-5 

3-0 


917 


237 


241 


246 


251 


256 


260 


265 


270 


275 


279 






918 


284 


289 


293 


298 


303 


308 


312 


317 


322 


327 


•7 
.8 
• Q 


3-5 


919 
920 

921 


331 


336 


341 


345 


350 


355 


360 


364 


369 


374 


4.0 

.1 . c. 


379 3^ 


388 


393 


397 


402 


407 


412 


416 


421 




426 


430 


435 


440 


445 


449 


454 


459 


463 


4^8 


922 


473 


478 


482 


487 


492 


496 


501 


506 


511 


515 




923 


520 


525 


529 


534 


539 


543 


548 


553 


558 


562 




924 


567 


572 


576 


58i 


586 


590 


595 


600 


605 


609 




925 


614 


619 


623 


628 


^33 


637 


642 


647 


651 


656 




926 


661 


666 


670 


675 


680 


684 


689 


694 


^^98 


703 




927 


708 


712 


717 


722 


726 


731 


736 


741 


745 


750 




928 


755 


759 


764 


769 


773 


778 


783 


787 


792 


797 




929 
930 

931 


801 


806 


811 


815 


820 


825 


829 


834 


839 


843 


^ 


^48 
S95 


853 


857 


862 


867 


871 


876 


881 


885 


896 


899 


904 


909 


913 


918 


923 


927 


932 


937 




4 


932 


941 


946 


951 


955 


960 


965 


969 


974 


979 


983 


. I 



0.4 


933 


^88 


993 


997 


^^002 


*oo7 


*OII 


*oi6 


*020 


*025 


^030 


•3 


u 9 
1.3 


934 


97 034 


039 


044 


048 


053 


058 


062 


067 


072 


076 






935 


081 


086 


090 


095 


099 


104 


109 


113 


118 


123 


.4 


1.8 


936 


127 


132 


137 


141 


146 


151 


155 


160 


164 


169 


•5 
.6 


2.7 


937 


i74 


178 


183 


188 


192 


197 


202 


206 


21 1 


215 






93« 


220 


225 


229 


234 


239 


243 


248 


252 


257 


262 


•7 
.8 

• 9 


31 

a. 6 


939 
940 

941 


265 
3^3 


271 

317 


276 


280 


285 


289 
33^ 


294 
340 


299 

345 


303 


308 


322 


328 


33"^ 

377 


349 
396 


354 




359 


3^3 


368 


373 


382 


386 


391 


406 


942 


405 


409 


414 


419 


423 


428 


432 


437 


442 


446 




943 


451 


456 


460 


465 


469 


474 


479 


483 


488 


492 




944 


497 


502 


506 


511 


515 


520 


525 


529 


534 


538 




945 


543 


548 


552 


557 


561 


566 


570 


575 


580 


584 




946 


589 


593 


598 


603 


607 


61 2 


616 


621 


626 


630 




947 


635 


639 


644 


649 


653 


658 


662 


667 


671 


676 




948 


681 


685 


690 


694 


699 


703 


708 


713 


717 


722 




949 
950 


725 

772 


731 


736 


740 


745 


749 


754 


758 


763 


768 




777 


781 


786 


790 


795 


800 


804 


809 


813 


N. 





1 


2 


3 


4 


5 


6 


7 


8 





P. P. 



341 







TABLE 


v.— LOGARITHMS OF 


NUMBERS 








' N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


950 


97 772 
818 


777 
822 


781 


786 


796 


795 


800 


804 


809 


^^3 


, 


951 


827 


83 J 


836 


841 


845 


850 


854 


859 


952 


S63 


868 


873 


877 


882 


886 


891 


895 


900 


904 




953 


909 


914 


918 


923 


927 


932 


936 


941 


945 


950 




954 


955 


959 


964 


968 


973 


977 


982 


986 


991 


996 




1 955 


98 000 


005 


009 


014 


oig 


023 


027 


032 


036 


041 


5 


956 


046 


050 


055 


059 


064 


068 


073 


077 


082 


086 


.1 


0.5 


957 


091 


095 


100 


105 


109 


114 


118 


123 


127 


132 


.2 


I.O 


958 


136 


141 


145 


150 


154 


159 


163 


168 


173 


177 


•3 


1-5 


959 
960 

961 


182 
227 
272 


186 


191 


195 


200 


204 


209 


213 


218 


222 


•4 

•5 
.6 


2.0 

2.5 
3-0 


231 


236 


246 


245 


249 


254 


259 


263 


268 


277 


281 


286 


296 


295 


299 


304 


308 


313 


962 


317 


322 


326 


33^ 


335 


340 


344 


349 


353 


358 


■ 7 


3-5 


1 963 


362 


367 


371 


376 


380 


385 


389 


394 


398 


403 


.8 


4.0 


! 964 


407 


412 


416 


421 


425 


430 


434 


439 


443 


448 


•9 


4-5 


965 


452 


457 


461 


466 


470 


475 


479 


484 


488 


493 




966 


497 


502 


506 


511 


515 


520 


524 


529 


533 


538 




967 


545 


547 


551 


556 


560 


565 


569 


574 


578 


583 




968 


587 


592 


596 


601 


605 


610 


614 


619 


623 


628 




969 
970 

971 


632 

677 

722 


637 


641 


646 


650 


655 


659 


663 


668 


672 


, 4 


681 


686 


696 


695 


699 


704 


708 


713 


717 


726 


731 


735 


740 


744 


749 


753 


757 


762 


.1 


0.4 


972 


766 


771 


775 


780 


784 


789 


793 


798 


802 


807 


.2 


0.9 


973 


8ii 


815 


820 


824 


829 


^33 


838 


842 


847 


85? 


•3 


1-3 


974 


856 


860 


865 


869 


873 


878 


882 


887 


891 


896 


•4 


1.8 


975 


900 


905 


909 


914 


918 


922 


927 


931 


936 


940 


•5 
.6 


2.2 


976 


945 


949 


954 


958 


963 


967 


971 


976 


980 


985 


2.7 


977 


989 


994 


998 


*oo3 


*oo7 


*OII 


*oi6 


*020 


*025 


*029 


• 7 


3-1 


' 978 


99 034 


038 


043 


047 


051 


056 


060 


065 


069 


074 


.8 


3.6 


i 979 
, 980 

1 981 


078 


082 


087 


091 


096 


100 


105 


109 


113 


118 


= 9 


4.0 


122 


127 


131 


136 


146 


145 


149 


153 


158 


162 




167 


171 


176 


180 


184 


189 


193 


198 


202 


206 


982 


211 


215 


220 


224 


229 


233 


237 


242 


246 


251 




983 


255 


260 


264 


263 


273 


277 


282 


286 


290 


295 




984 


299 


304 


308 


312 


317 


321 


326 


330 


335 


339 




985 


343 


348 


352 


357 


361 


365 


370 


374 


379 


383 


4 


986 


387 


392 


396 


401 


405 


409 


414 


418 


423 


427 


.1 


0.4 


i 987 


431 


436 


440 


445 


449 


453 


458 


462 


467 


471 


.2 


0.8 


1 988 


475 


480 


484 


489 


493 


497 


502 


506 


511 


515 


•3 


1 .2 


989 
990 

991 


519 

563 
607 


524 


528 


533 


537 


541 


54^ 


550 


554 


559 


•4 

•5 
.6 


1.6 
2.0 
2.4 


568 


572 


576 


581 


585 


590 


594 


598 


603 


611 


616 


626 


625 


629 


^33 


638 


642 


647 


992 


651 


655 


660 


664 


668 


673 


677 


682 


686 


696 


•7 


2.8 


993 


695 


699 


703 


708 


712 


717 


721 


725 


730 


734 


.8 


3.2 
3.6 


994 


738 


743 


747 


751 


756 


760 


765 


769 


773 


778 


•9 


i 995 


782 


786 


791 


795 


800 


804 


808 


813 


817 


821 




996 


826 


836 


834 


839 


843 


847 


852 


856 


861 


865 




997 


869 


874 


878 


882 


887 


891 


895 


900 


904 


908 




998 


913 


917 


922 


926 


93^ 


935 


939 


943 


948 


952 




999 
1000 


956 

00 000 


961 


965 


969 


974 


978 


982 


987 


991 


995 




004 


oog 


013 


017 


021 


026 


030 


034 


039 


1 ^• 





1 


2 


3 


4 


5 


G 


7 


8 


9 P. P. 1 



342 







TABLE 


v.— LOGARITHMS OF 


NUMBERS 


• 






• 


N. 





1 


2 1 3 1 4 


5 1 6 


7 8 1) 


P. P 


. 


1000 

OI 


000 000 
434 


043 


087 1 136 


J 73 
607 


217 266 


304 


347 


390 






477 


521 


564 


651 


694 


737 


781 


824 


02 


867 


911 


954 


997 


*o4i 


*o84 


*I27 


*i7i 


^214 


*257 






03 


001 301 


344 


387 


431 


474 


517 


566 


604 


647 


696 






04 


733 


777 


820 


863 


906 


950 


993 


*036 


*o79 


*I23 






'^5 


002 166 


209 


252 


295 


339 


382 


425 


468 


511 


555 






c6 


59S 


641 


684 


727 


770 


814 


857 


900 


943 


9^'^6 






07 


003 029 


072 


115 


159 


202 


245 


288 


331 


374 


417 


47 


43 


oS 


466 


503 


545 


590 


^33 


676 


719 


762 


805 


848 


.1 


4-3 


4-3 


09 
1010 

1 1 


891 


934 


977 


*026 


^063 
493 


*io6 
536 


*i49 
579 


*I92 

622 


*235 
665 


*278 


.2 

•3 

•4 
• 5 

.6 


8.7 
13.6 

174 
21.7 
26. 1 


8.6 
12.9 1 

17.2 

25.8 I 


004 321 
751 


364 


407 


450 


708 


794 


837 


880 


923 


966 


*oo9 


*o5i 


*094 


*i37 


12 


005 186 


223 


265 


309 


352 


395 


438 


481 


523 566 








13 


609 


652 


695 


738 


781 


824 


866 


909 


952 ! 995 


•7 
.8 


30 -4 
34.8 


30.1 
34-4 


14 


006 038 


081 


123 165 


209 


252 


295 


337 


386 423 


•9 


39- 1 


38.7 


15 


a66 


509 


551 i 594 


637 


680 


722 


765 


808 851 






16 


893 


936 


979 i*02 2 


^064 


*io7 


*i5o 


*i93 


*235 


*278 






17 


007 321 


3^3 


4O6 


449 


491 


534 


577 


620 


662 


705 






18 


748 


790 


S33 875 


918 


961 


*oo3 


*o46 


*o89 131 






19 

1020 

21 


008 174 
600 

009 025 


217 
642 

oog 


259 302 


344 
770 
196 


387 
813 


430 


472 


515 557 






685 


728 
153 


855 


898 


946 : 9S3 


III 


238 


281 323 


366 1 408 


22 


451 


493 


536 


578 


621 


663 


706 748 


790 i 833 


42 


42 


23 


875 


918 


966 


*oo3 


*045 


*o88 


*i36 


*I72 


""215 


*257 


.2 


4-^ 

8.5 


4.2 

8.4 


24 


010 300 


342 


385 


427 


469 


512 


554 


1 596 


639 


681 


■3 


12.7 


12.6 


25 


724 


766 


808 


851 


893 


935 


978 


*026 


*o62 


*io5 


■4 


17.0 
21 .2 


16.8 
21 .0 


26 


on 147 


189 


232 


274 


316 


359 


401 


443 


486 


528 


.6 


25-5 


25.2 


27 


570 


612 


655 


697 


739 


782 


824 


866 


908 


951 


•7 


29-7 


29.4 


28 


993 


*o35 


*o77 


*I20 


*l62 


*204 


*246 


*288 


*33^ 


*373 


.8 
•9 


340 
38.2 


33-6 
37-8 


29 
1030 

31 


012 415 
837 


457 


500 


542 


584 


625 
^048 


668 


710 


753 


795 






879 


921 


963 i*oo6 


*OQO 
511 


*I32 


174 


216 

637 


013 258 


301 


343 


385 


427 


469 


553 


595 


32 


679 


722 


764 


806 


848 


890 


932 


974 


*oi6 


*o58 






33 


014 100 


142 


184 


225 


263 


310 


352 


394 


436 


478 






34 


526 


562 


604 


648 


688 


730 


772 


814 


856 


898 






i 35 


940 


982 


*024 


*o66 


*io8 


*i50 


*I92 


*234 


*276 


*3i8 






1 3'^ 


015 360 


401 


443 


485 


527 


569 


611 


653 


695 


737 


0S 

Al 


41 


' 37 


779 


820 


862 


904 


946 


988 


^030 


*072 


*TI3 


155 


.1 


41 


4.1 
9, 1 


3^ 


016 197 


239 


281 


323 


364 


406 


448 


490 


532 


573 


■3 


12.4 


12.3 


39 
1010 

41 


615 

017 033 

450 


657 


699 


741 


782 


824 


866 


908 


950 


991 


•4 

•5 
.6 


16.6 
20.7 
24.9 


16.4 

20.5 . 

24.6 , 


075 
492 


117 

534 


158 


206 


242 


284 


325 

742 


367 


409 
826 


576 


617 


659 


701 


784 


42 


867 


909 


95 J 


992 


*034 


*o76 


*ii7 


*i59 


*20I 


^242 


.8 


33-2 


2fc.7 
?2.8 


43 


01 S 284 


326 


367 


409 


451 


492 


534 


575 


617 


659 


•y 


37-3 


36.9 


1 44 


706 


742 


783 


825 


867 


908 


950 


991 


*os3 


*o74 






! 45 


019 116 


158 


199 


241 


282 


324 


365 


407 


448 


490 






1 46 


531 


573 


614 


656 


697 


739 


786 


822 


863 


905 






47 


946 


988 


*029 


*o7i 


*II2 


*i54 


*i95 


*237 


*278 


*320 






48 


020 361 


402 


444 


485 


527 


5^8 


610 


651 


692 


734 






49 

1050 


775 
021 189 


817 
236 

1 


858 


899 


941 


982 
396 


*024 

437 


*o65 *i06 


♦148 






272 
2 


3^3 354 


478 520 ' 561 


N. 


3 1 4 


r> <> 7 8 5) 


P. P 


• 



343 







TABLE 


v.— LOGARITHMS OF 


NUMBERS 


• 






i ^. 





1 


2 


3 


4 


5 ! 6 ! 7 I 8 


9 


P.P. 


1050 

51 


021 189 
602 


230 


272 


313 


3^)4 


396 


437 


478 
892 


520 


561 




644 


685 


726 


'16?, 


809 


856 


933 


974 


52 


022 015 


057 


098 


139 


181 


222 


263 


304 


346 


387 


41 


53 


423 


469 


511 


552 


593 


634 


676 


717 


758 


799 


.1 


4.1 

8 -i 


54 


840 


882 


923 


964 


*oo5 


*o46 


*o88 


*I29 


*i7o 


*2II 


•3 


12.4 


: 55 


023 252 


293 


zzs 


376 


417 


458 


499 


540 


58i 


623 


•4 


16.6 


! 56 


664 


705 


746 


787 


828 


869 


910 


951 


993 


*o34 


•5 

.6 


20.7 
24.9 


' 57 


024 075 


116 


157 


198 


239 


280 


321 


362 


403 


444 






5^ 


485 


526 


568 


609 


650 


691 


732 


773 


814 


855 


.8 


33-2 


59 

1060 

61 


896 
025 306 

715 


937 


978 


*oi9 


*o6o 


*IOI 


^142 


*i83 


*224 


*265 


•9 


37.3 


347 


388 


429 


469 


5TO 


551 


592 


^3% 


674 

*o83 


1 


756 


797 


^Z^ 


879 


920 


961 


*002 


^042 


1 62 


026 124 


165 


205 


247 


288 


329 


370 


410 


451 


492 


.41 1 


63 


533 


574 


615 


656 


696 


737 


778 


819 


860 


901 


.1 
.2 


4.1 
8.2 


64 


941 


982 


*023 


*o64 


*io5 


*i45 


*i86 


*227 


*268 


*309 


•3 


12.3 


65 


027 349 


390 


431 


472 


512 


553 


594 


635 


675 


7^6 


.4 


16.4 


66 


757 


798 


^z^^ 


879 


920 


961 


*OOI 


*042 


*o83 


*I23 


.6 


24.6 


1 67 


028 164 


205 


246 


285 


327 


368 


408 


449 


490 


530 


•7 


28.7 


1 68 


571 


612 


652 


693 


734 


774 


815 


856 


896 


937 


.8 
■ Q 


32.8 

26. Q 


69 
1070 

1 71 


977 
029 384 

789 


*oi8 


*o59 


*099 


^140 


*i8i 


*22I 


^262 


*302 


*343 


J 


424 


465 


505 


546 


586 


627 


668 


708 


749 


830 


870 


911 


951 


992 


*032 


*o73 


*ii4 


*i54 


72 


030 195 


235 


276 


2>^6 


357 


397 


438 


478 


519 


559 


.1 


4.0 


73 


599 


040 


680 


721 


761 


802 


842 


883 


923 


964 


.2 
.•1 


8.1 
12. 1 


74 


031 004 


044 


085 


I2g 


166 


205 


247 


287 


327 


368 






75 


408 


449 


489 


529 


570 


610 


651 


691 


73? 


772 


•4 
•5 


20.2 


76 


812 


852 


893 


933 


973 


*oi4 


*o54 


*094 


""^ZS 


*i75 


.6 


24-3 


77 


032 215 


256 


296 


3Z6 


377 


417 


457 


498 


538 


578 


•7 

.8 


28.3 
^2.4 


^ 78 


619 


659 


699 


739 


780 


820 


866 


900 


941 


981 


•9 


36.4 


1 79 
1080 

' 81 


033 021 

424 
825 


061 


102 


142 


182 


222 


263 


303 


343 


383 


Ar\ 


464 


504 


544 


584 


625 


665 


705 


745 


785 


866 


906 


946 


986 


*025 


*o66 


*io7 


147 


187 


82 


034 227 


267 


307 


347 


388 


428 


468 


508 


548 


588 


.1 


4.0 


1 ^2 


623 


668 


708 


748 


789 


829 


869 


909 


949 989 


.2 
.3 


8.0 

12.0 


' 84 


035 029 


069 


109 


149 


189 


229 


269 


309 


349 


3S9 


.4 


16.0 


1 85 


429 


470 


510 


550 


590 


630 


670 


710 


750 


790 


•S' 


20.0 


86 


830 


870 


910 


950 


990 


^029 


^069 


*i09 


*i49 


*i89 


.6 


24.0 


87 


036 229 


269 


309 


349 


389 


429 


469 


509 


549 


589 


•7 

.8 


28.0 
32.0 


88 


629 


669 


708 


748 


788 


828 


868 


908 


948 


988 


.9 


36.0 


89 
1090 

91 


037 028 


068 


107 


147 


187 


227 


267 


307 

705 


347 


386 


!^0 I 


425 

825 


465 


506 


546 


586 


625 


665 


745 ! 785 


864 


904 


944 


984 


*023 


^063 


*io3 


143 


183 


92 


038 222 


262 


302 


342 


38J 


421 


461 


501 


540 


580 


.1 


3.§ 


92, 


620 


660 


699 


739 


779 


819 


858 


898 


938 


977 


.2 
.3 


7-9 
11-8 


94 


039 017 


057 


096 


^ze 


176 


216 


255 


295 


*^ /I •* 


374 


•4 


15-8 


95 


414 


454 


493 


533 


572 


612 


652 


691 


73 i 


771 


•5 


19.7 


96 


810 


850 


890 


929 


969 


*oo8 


^048 


*o88 


*I27 


*i67 




23-/ 


97 


040 205 


246 


286 


325 


365 


404 


444 


483 


523 


5^3 


•7 
.8 


27-6 ' 
31-6 


98 


602 


642 


681 


721 


766 


800 


839 


879 


918 


958 


•9 


35-5 1 


99 
1100 


997 
041 392 


*o37 


*o76 


*ii6 


*i55 


*i95 


*234 


*274 


^Z^l ""353 




432 


471 


511 


550 


590 


629 


669 


708 748 


N. 





1 


2 


3 


4 


5 


6 


7 


8 9 


P.P. 



344 



VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



log sin ip = log (p" -\- S. 




0° 


log 


' = log sin (p -\- S'. 1 
= log tan 04- 7". II 




log tan (p = log ' + T. 






log (p 




'' 


( 


S 
4.685 57 


T 


Lo^. Sin. 


S' 


T 


LoJT. Tan. | 




o 





57 


— 00 


5.31442 


42 


— 00 t 




60 


I 


57 


57 


6.46 372 


42 


42 


6.46 372 




120 


2 


sf 


57 


.76475 


42 


42 


.76475 




180 


3 


Si 


Si 


.94 084 


42 


42 


.94 084 




240 


4 


Si 


Si 


7.06 578 
7.16 269 


42 


42 


7.06 578 
7.16 269 




300 


5 


4.685 Si 


57 


5.31442 


42 




360 


6 


Si 


Si 


.24187 


42 


42 


.24188 




420 


7 


57 


Si 


.30 882 


42 


42 


.30882 




480 


8 


Si 


57 


.36681 


42 


42 


.36681 




540 
600 


9 
10 


Si 


Si 


.41 797 


42 


42 


.41 797 




4.685 si 


Si 


7.46 372 


5.31442 


42 


7.46 372 




660 


II 


57 


Si 


.50512 


42 


42 


.50512 




720 


12 


Si 


Si 


.54 290 


42 


42 


.54291 




780 


13 


Si 


57 


• 57 767 


42 


42 


• 57 767 




840 


14 


Si 


57 


.60985 


42 


42 


.60 985 




900 


15 


4.685 57 


58 


7.63 981 


5.31442 


42 


7.63 982 




960 


16 


Si 


58 


.66 784 


42 


42 


.66 785 




1020 


17 


57 


58 


.6941^ 


42 


42 


.69418 1 




1080 


18 


57 


58 


.71 899 


42 


42 


.71 900 1 




1 140 


19 


57 
4.685 57 


58 


_ ^.74.248 


42 


42 


.74 248 ' 




1200 


20 


58 


7.76475 


5.31443 


42 


7.76476 




1260 


21 


57 


58 


.78 594 


43 


42 


.78 595 




1320 


22 


57 


58 


.80614 


43 


42 


.80615 




1380 


23 


57 


58 


.82 545 


43 


42 


.82 546 




1440 


24 


57 


58 


•84 393 


43 


42 


•84 394 




1500 


25 


4.685 57 


58 


7.86 166 


5.31443 


41 


7.86 167 




1560 


26 


57 


58 


.87 869 


43 


41 


.87871 




1620 


27 


57 


58 


.89 508 


43 


41 


.89 510 




1680 


28 


57 


58 


.91 088 


43 


41 


.91 089 




1740 


29 


57 


58 


.92 612 


43 


41 


.92613 




1800 


30 


4.685 57 


58 


7.94 084 


5.31443 


41 


7.94 086 




i860 


31 


57 


58 


•95 508 


43 


41 


.95510 




1920 


32 


57 


58 


.96 887 


43 


41 


.96 889 




1980 


33 


57 


59 


.98 223 


43 


41 


.98 225 




2040 


34 


57 


59 


.99 520 


43 


41 


•99 522 




2100 


35 


4.685 56 


59 


8.00 778 


5.31443 


41 


8.00781 




2160 


36 


56 


59 


.02 002 


43 


41 


.02 004 




2220 


37 


56 


59 


.03 192 


43 


4J 


.03 194 




2280 


38 


56 


59 


.04 350 


43 


40 


•04352 




2340 


39 


56 
4.685 56 


59 


.05 478 


43 


40 


.05481 




2400 


40 


59 


8.06 577 


5.31443 


46 


8.06 580 




2460 


41 


56 


S9 


.07 650 


43 


40 


.07 653 




2520 


42 


56 


59 


.08 695 


43 


40 


.08 699 




2580 


43 


56 


60 


.09718 


43 


40 


.09721 




2640 


44 


56 


60 


.10716 


43 


40 


.10720 




2700 


45 


4.685 56 


60 


8. 1 1 692 


5.31444 


40 


8. 1 1 696 




2760 


46 


56 


60 


.12647 


44 


40 


.12 651 




2820 


47 


56 


60 


.13581 


44 


40 


.13585 




2880 


48 


56 


60 


.14495 


44 


39 


.14499 




^940 


49 


56 


60 


.15390 


44 


39 


•15395 




3000 


50 


4.685 56 


60 


8.16268 


5.31444 


39 


8.16272 




3060 


51 


56 


60 


.17 128 


44 


39 


• 17 133 




3120 


52 


56 


61 


.17971 


44 


39 


• 17 976 




3180 


53 


56 


61 


.18798 


44 


39 


.18803 




3240 


54 


55 


61 


.19610 


44 


39 


.19615 




3300 


55 


4.685 Si 


61 


8.20 407 


5-3H44 


39 


8.20412 




3360 


56 


Si 


61 


.21 189 


44 


38 


.21 195 




3420 


57 


55 


61 


.21958 


44 


38 


.21 964 




3480 


S« 


55 


61 


.22713 


44 


38 


.22719 




3540 


_ 59_ 


si 


62 


.23 45J 


44 


38 


J ^23 462 _ 





345 



TABLE 


VI.— LOGARITHMIC SINES AND TANGENTS OF 


SMALL ANGLES.! 


log sin if) =z log 0" -\- S. 10 log (p" = log sin -f- 5". 1 
log tan cp = log <P" + ^- ^ log <P" = log tan -(- 7", | 


// 


r 


S 


T 


Log. Sin. 


S' 


T 


Log. Tan. 


3600 
3660 
3720 
3780 
3840 




I 

2 

3 

4 


4.685 55 

55 
55 
55 
55 


62 
62 
62 
62 
62 


8.24185 

.24 903 
.25 609 
.26 304 
.26 988 


5-31444 
45 
45 

45 
45 


38 
38 
38 
37 
37 


8.24 192 
.24910 
.25616 
.26 31? 
-26 99§ 


3900 
! 3960 
4020 
4080 
4140 


5 
6 

7 
8 

9 


4.685 55 

55 
54 
54 
54 


62 
63 
63 
63 
63 


8.27 661 
.28 324 
.28 97 f 
.29 626 
.30254 


5-31445 

45 
45 

45 . 
45 


37 
37 
37 
37 
36 


8.27 669 
.28 332 
.28985 
.29629 
.30 263 


4200 
4260 
4320 
4380 
4440 


10 

II 
12 

J3 
14 


4.685 54 
54 
54 
54 
54 


63 
63 
64 
64 
64 


8.30 879 

.31495 
.32 102 
.32 701 
•33 292 


5-31445 
45 
45 

46 
46 


36 
36 
36 
36 
36 


8.30 888 

.31 504 
.32 112 
.32711 
.33 302 


4500 
4560 
1 4620 
4680 
4740 


15 
16 

17 
18 

19 


4.685 54 
54 
54 
54 
53 


64 
64 
65 
65 
65 


8.33875 

.34450 
.35018 

■35 578 
.36131 


5.31446 

46 
46 
46 

46 


35 
35 
35 
35 

35 


8.33 885 
.34461 
.35029 
•35 589 
.36 143 


4800 
4860 
4920 
4980 
5040 


20 

21 

22 

23 

24 


4-685 53 
53 
53 
53 
53 


6$ 

65 
65 

66 
66 


8.36677 
.37217 

.37750 
.38 276 
.38 796 


5.31446 
46 
46 
46 
47 


34 
34 
34 
34 
34 


8.36689 
.37 229 
.37 762 
.38 289 
.38 809 


5100 
5160 
5220 
5280 
. 5340 


25 
26 

27 
28 

29 


4.685 53 
53 
53 

52 
52 


66 
66 
67 
67 
67 


8.39310 
.39818 
.40 320 
.40816 
•41 307 


5-31447 
47 
47 
4? 
4f 


33 
33 
33 
33 
33 


8.39 323 
.39 831 
•40 334 
.40 836 
.41 321 


5.400 
5460 
5520 
5580 
5640 . 


30 

31 

32 
33 
34 


4.685 52 

52 

52 

52 
52 


67 
67 
68 
68 
68 


8.41 792 
.42 271 
.42 746 

.43215 
.43 680 


5-3144^ 

4l 
48 

48 


32 
32 
32 
32 
31 


8.41 807 
.42 287 
.42 762 
.43 231 
.43 696 


5700 
5760 
5820 
5880 
5940 


35 
36 

37 
38 
39 


4.685 52 
52 
51 
51 
51 


68 
69 

69 
69 
69 


8.44139 
.44 594 
.45 044 
.45 489 

•45 930 


5.31448 
48 
48 
48 
48 


31 
31 
31 
30 
30 


8.44 1 56 
.44611 
.45 061 
.45 507 
.45 948 


i 6000 
6060 
6120 
6180 
6240 


40 

41 

42 

43 
44 


4.685 51 

51 
51 
51 

51 


69 
70 

70 
76 
70 


8.46 366 
.46 798 
.47 226 
.47 650 
.48 069 


5.31448 
49 
49 
49 

49 


30 
30 
30 
29 
29 


8.46 385 
.46817 

•47 245 

• .47 669 

.48 089 


6300 
6360 
6420 
6480 
6540 


45 
46 

47 
48 

49 


4.685 56 

50 
50 
55 
50 


71 

71 
71 

72 
72 


8.48 485 
.48 896 
.49 304 
.49 70S 
.50 108 


5-31449 
49 
49 
49 
50 


29 

28 
28 

28 
28 


8.48 505 
.48917 

•49325 
.49 729 
.50 130 


6600 
6660 
6720 
6780 
6840 


50 

51 

52 
53 

54 


4.685 50 
SO 
50 
49 
49 


72 
72 

73 
73 
73 


8.50 504 
•50897 
.51 286 
.51 672 
.52055 


5-31450 
50 
50 

55 
53 


2? 
2j 

27 

27 

26 


8.50 526 
.50920 
.51 310 
.51 696 
.52079 


6900 
6960 
7020 
7080 
7140 


55 
56 
57 
58 
59 


4.685 49 

49 
49 
49 
49 


73 
74 
74 
74 
75 


8.52434 
.52 810 

.53183 
•53552 
.53918 


5-31450 
51 
51 
51 
51 


26 

26 

25 
25 
25 


8.52458 
.52835 
.53 208 
-53578 
•53 944 



346 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



log sin 


:/j = log (p'' -f S. 




2° 


log = lug 


sin ip-\- S'. 


log tan < 


p = log <p" + r. 




log 0" = log 


tan 04- 7". 


" 


' 


S 


T 


Lo^. Sill. 


S' 


T 


Lojr. Tan. 


, 7200 





4.685 48 


75 


8.54282 


5-3«4 5i 


25 


8-54 308 


1 7260 


I 


48 


75 


.54642 


51 


24 


.54669 


7320 


2 


48 


75 


•54 999 


51 


24 


•55027 


7380 


3 


48 


76 


.55354 


52 


24 


.55381 


1 7440 


4 


48 


76 


•55705 


52 


23 


•55 733 
8.56083 


7500 


5 


4.68548 


76 


8.56054 


5.31452 


23 


' 7560 


6 


48 


77 


.56 400 


52 


23 


.56429 


7620 


7 


47 


77 


• 56743 


52 


22 


•56772 


7680 


8 


47 


77 


• 57083 


52 


22 


•57 113 1 


7740 


9 


47 


78 


.57421 


52 


22 


•57452 i 


, 7800 


10 


4.685 47 


78 


8^57 756 


5-3U53 


22 


8.57 787 


7860 


1 1 


47 


78 


.58 089 


53 


21 


.58 121 


7920 


12 


47 


79 


.58419 


53 


21 


-58 45t 


7980 


13 


46 


79 


•58 747 


53 


21 


-58779 


8040 
8100 


14 


46 


79 


.59072 


53 


26 


.59105 


15 


4.685 46 


80 


8.59395 


5-31453 


20 


8-59 428 


8160 


16 


46 


80 


•59715 


54 


20 


-59 749 


8220 


17 


46 


86 


.60 033 


54 


19 


.60 067 


1 8280 


18 


46 


81 


.60 349 


54 


19 


.60 384 


; 8340 


19 


45 


81 


.60 662 


54 


19 
18 


.60 698 1 
8.61 009 1 


1 8400 


20 


4.68545 


81 


8.60 973 


5.31454 


1 8460 


21 


45 


82 


.61 282 


54 


18 


.61 319 


8520 


22 


45 


82 


.61 589 


55 


18 


.61 626 


8580 


23 


45 


82 


.61 893 


55 


I? 


.61 931 


8640 


24 


45 


83 


.62 196 


55 


17 


•62 234 


8700 


25 


4.68544 


83 


8.62 496 


5-31455 


16 


8.62 535 


8760 


26 


44 


83 


•62 795 


55 


16 


.62 834 


8820 


27 


44 


84 


.63 091 


55 


16 


•63 131 


8880 


28 


. 44 


84 


•63 385 


56 


j5 


•63425 


8940 


29 


44 


84 


•63 677 


56 


15 


•63 7 18 


9000 


30 


4.68543 


85 


8.63 968 


5-314 56 


15 


8..64 009 


9060 


31 


43 


8^ 


.64256 


56 


14 


.64 298 


9120 


32 


43 


86 


•64 543 


56 


14 


.64 585 


9180 


33 


43 


86 


.64 827 


57 


14 


.64 870 


9240 


34 


43 


86 


.65 1 10 


57 


13 


•65153 


9300 


35 


4.68543 


87 


8.65 391 


5-3H57 


13 


8.65 435 


9360 


36 


42 


87 


.65 670 


57 


12 


.65715 


9420 


37 


42 


87 


•65 947 


57 


12 


■^S 993 


9480 


33 


42 


88 


.66 223 


58 


12 


.66 269 


9540 
9600 


39 


42 


88 


.66 497 


58 


II 


.66 543 


40 


4.685 42 


89 


8.66 769 


5-31458 


II 


8.66816 


9660 


41 


41 


89 


.67 039 


58 


10 


.67 0S7 


9720 


42 


41 


89 


.67 308 


58 


10 


•67 356 


9780 


43 


41 


90 


.67575 


59 


10 


.67 624 


9840 


44 


41 


90 


.67 845 


59 


09 


.67 890 


9900 


45 


4.685 41 


91 


8.68 104 


5^31459 


09 


8.68 I 54 


9960 


46 


40 


91 


.68 366 


59 


08 


.68417 


10020 


47 


40 


91 


.68 627 


59 


08 


.68 678 


10080 


48 


40 


92 


.68 886 


60 


08 


.68 938 


10140 


49 


40 


92 


.69144 


60 


of 


•69 '96 


10200 


50 


4.685 40 


93 


8.69 400 


5.31460 


07 


8.69453 


10260 


51 


39 


93 


.69654 


60 


06 


.69 708 ' 


10320 


52 


39 


93 


.69 907 


66 


06 


.69 96 ! 


10380 


53 


39 


94 


.70159 


61 


06 


.70214 


10440 


54 


39 


94 


.70 409 


61 


05 


.70 464 


10500 


55 


4.685 38 


95 


8.70657 


5.31461 


05 


8.70714 


10560 


56 


38 


95 


.70905 


61 


04 


.70 962 


10620 


57 


38 


96 


.71 150 


61 


04 


.71 208 


10680 


58 


38 


96 


.71 395 


62 


03 


•71 453 


10740 


59 


38 


97 


•71638 


62 


03 


.71697 



347 



TABLE VIL— LOGARITHMIC 



SINES, COSINES, 



TANGENTS, AND COTANGENTS 



/ 


Log. Sin. 


D 


Log. Tan. 


Com. D. 


Log. Cot. 


Log. Cos. 







— 00 




— 00 




-i- 00 


0.00000 


GO 


I 


6.46 372 




6.46 372 




3-5362^ 


0.00000 


59 


2 


e>.'j6Ail 


17609 


6.76475 


17609 


3.23 524 


0.00000 


58 


3 


6.94084 


6.94084 


3-05915 


0.00000 


57 


4 


7.06 578 


12494 
9691 
7918 
6695 


7.06 578 


12494 
9691 


2.93421 


0.00000 


56 


5 


7.16 269 


7.16 269 


2.83736 


0.00 000 


55 


6 


7.24 1 8f 


7.24188 


791 8 


2.75 812 


0.00000 


54 


7 


7.30882 


7.30882 


6094 


2.69 11^ 


0.00000 


53 


8 


7.36681 


5799 


7.36681 


5799 


2.63 3I8 


0.00000 


52 


9 


7.41 797 


5"5 

4575 


7.41 797 


5"5 
4575 
4139 


2.58 203 


0.00000 


51 


10 


7.46 372 


7.46372 


2.5362^ 


0.00 000 


50 


II 


7.50512 


3778 


7.50512 


2.49488 


0.00000 


49 


12 


7.54290 


7.54291 




2.45 709 


9.99999 


48 


13 


7.57767 


3476 


7.57767 


3476 


2.42 233 


9.99999 


47 


14 


7.60985 


3218 
2996 
2803 
2633 


7.60985 


3218 
2996 
2803 


2.39014 


9.99999 


4.6 


15 


7.63981 


7.63982 


2,36018 


9.99999 


45 


i6 


7.66784: 


7.66 785 


2.33215 


9.99999 


44 


17 


7.6941^ 


7.69418 


2633 


2.30 582 


9.99999 


43 


i8 


7.71 899 




7.71 900 


2482 


2.28 099 


9.99999 


42 


19 


7.74248 


2348 
2227 

2119 


7.74248 


2348 
2227 


2.25751 


9.99999 


41 


20 


7.76475 


7.76476 


2.23 524 


9.99999 


40 


21 


T.^Z 594 


I'l^ 595 




2.21 405 


9.99999 


39 


22 


7.80614 




7.80615 




2.19384 


9.99999 


38 


23 


7.82545 


1930 


7.82 546 


1930 


2.17454 


9.99999 


37 


24 


7.84393 


1843 
1772 


7-84394 


1848 
1773 


2.15 605 


9.99999 


36 


25 


7.86 166 


7.86 16^ 


2.13832 


9.99999 


35 


26 


7.87869 


1703 
1639 


7.87871 


1703 
1639 


2.12 129 


9.99999 


34 


27 


7.89 508 


7.89510 


2.10490 


9-99 998 


33 


28 


7.91 088 


1579 


7.91 089 


1579 


2.08 916 


9-99 998 


32 


29 


7.92 612 


1524 


7.92613 


1524 


2.07 386 


9-99 998 


31 


30 


7.94084 


1472 


7.94086 


1472 
1424 


2.05 914 


9-99 998 


30 


3i 


7.95 508 




7.95510 


2.04 490 


9.99998 


29 


32 


7.96 887 


1379 


7.96 8S9 


1379 


2.03 III 


9-99998 


28 


33 


7.98 223 


1336 


7.98225 


1336 


2.01 774 


9.99998 


27 


34 


7.99 520 


1296 


7.99 522 


1296 


2.00478 


9.99998 


26 


1 35 


8.00 778 


1253 
1223 
1 190 
1158 
1128 
1099 
1072 


8.00781 


■1^59 
1223 
iigo 
1158 


1.99 219 


9.9999? 


25 


i 36 
37 


8.02 002 

8.03 192 


8.02 004: 

8.03 191 


1.97995 
1.96 805 


9.9999? 
9.99997 


24 
23 


38 


8.04 350 


8.04 352 


1.95 647 


9.99997 


22 


39 


8.05 478 


8.05481 


1123 
1099 
1072 


1.94 519 


9.99997 


21 


40 


8.06 577 


8.06 580 


1-93 419 


9-99 997 


20 


41 


8.07 650 




8.07653 




1.92 347 


9.99997 


19 


42 


8.08 696 




8.08 699 




1. 91 300 


9-99 997 


18 


43 


8.09 718 


998 
976 


8.09721 




1.90278 


9-99 996 


17 


1 44 


8.10716 


8.10720 


999 
976 


1.89 279 


9-99 996 


16 


45 


8. 1 1 692 


8. 1 1 696 


1.88303 


9-99 996 


15 


46 


8.12647 




8.12651 


954 


1.87 349 


9.99996 


14 


47 


8.13 581 


934 


8.13585 


934 


1.86 415 


9.99996 


13 


48 


8.14495 


914 

895 
877 

860 


8.14499 


914 


1.85 506 


9.99996 


12 


49 


8.15396 


8.15395 


895 
877 
860 


1.84605 

1.8372? 


9-99 995 


II 


50 


8.16268 


8.16272 


9.99995 


10 


51 


8.17 128 


843 


8.17 133 


843 
827 


1.82867 


9-99 995 


9 


52 


8.17 971 


8.17 976 


1.82023 


9-99 995 


8 


53 


8.18798 


811 

797 
782 

768 


8.18803 


1. 81 igg 


9-99 995 


7 


54 


8.19610 


8.19615 


797 
783 
763 


1.80384 


9-99 994 


6 


55 
56 


8.20407 
8.21 189 


8.20412 
8.21 195 


1.79587 
1.78804 


9-99 994 
9-99 994 


5 
4 


57 


8.21 958 


8.21 964 


1.78036 


9.99994 


3 


58 


8.22 713 


755 


8.22 719 


755 


1.77 286 


9.99994 


2 


59 


8.23455 
8.24185 


742 
730 


8.23462 


742 
730 


1.76538 


9-99 993 


I 


60 


8.24 192 


1.75808 


9-99 993 





1 


Log. Cos. 


D 


Log. Cot. 


Com. D. 


Log. Tau. 


Log. Sin. 


/ 



89^ 



348 



TABLE VII. — LOGARITHMIC SIXES, COSINES, TANGENTS, AND COTANGENTS. 

1 



TiOK. sin. 






8.24 18S 


I 


8. 24 903 


o 


8.25 609 


3 


8.26 304 


4 


8.26988 


5 


8.27 661 


6 


8.28324 


7 


8.28977 


8 


8.29620 


9 


8.30254 


10 


8.30879 


1 II 


8.31495 


12 


8.32 102 


■ 13 


8.32 701 


14 


8.33292 


15 


8.33 87S 


i6 


8.34450 


17 


8.35018 


i8 


8-35 578 


19 


8.36 1 31 


20 


8.36677 


21 


S.37 2r7 


22 


8.37750 


23 


8.38276 


24 


8.38796 


25 


8.39310 


26 


8.39818 


27 


8.40320 


28 


8.40816 


29 


8.41 307 


30 


8.41 792 


31 


8.42 271 


32 


8.42 746 


33 


843215 


34 


8.43 680 


35 


8.44 139 


36 


8.44 594 


37 


8.45 044 


38 


8.45489 


39 


8.45930 


40 


8.46 365 


41 


8.46798 


42 


8.47 226 


43 


8.47 650 


44 


8.48069 


45 


8.48485 


46 


8.48 896 


47 


8.49 304 


48 


8.49 708 


49 


8.50 108 


1 50 


8. 50 504 


1 51 


8.50897 


52 


8.51286 


' 53 


8.51 672 


54 


8.52055 


1 55 


8.52434 


56 


8.52810 


! 57 


8.53183 


58 


8.53552 


59 


8-53918 


GO 


8.542S2 



Lot;. Cos. 



D 



718 
706 
694 
684 

673 
663 

653 
643 
634 
625 
616 
607 
599 
591 
583 
575 
567 
560 

553 
546 

539 
533 
526 
520 

514 

503 
502 
496 
491 
485 

479 
474 
469 
464 
459 
454 
450 

44S 
440 
436 
432 
428 
423 
419 

415 
411 
407 
404 
400 
396 
393 
389 
386 
382 
379 
375 
373 
369 
366 

363 
I> 



Loir, Tan. 



Com, I). 



8.24 192 
8.24910 

8.25 6I6 

8.26 311 

8.26 99^ 



8.27 669 

8.28 332 
8,28985 

8.29 629 
8.30263 



8,30888 

8.31 504 

8.32 112 
8.32711 
8.33302 



8,33885 
8,34461 
8.35029 

8.35 589 

8.36 143 



8.36 689 

8.37 229 
8.37 762 
8.38289 
8.38809 



8.39323 
8.39831 

8.40334 
8.40830 
8.41 321 



8.41 807 

8.42 287 

8.42 762 

8.43231 

8.43 696 



8.44 156 

8.44 611 
8.45061 

8.45 507 
8.45 948 



8.46 385 
8,46817 

8.47 245 
8.47 669 
8.48089 



8.48 505 
8.48917 
8.49325 

8.49729 
8,50130 



8-50526 

8.50 920 

8.51 310 
8,51 696 
8,52079 

'875^2"458~ 
8.52835 
8.53208 
8.53578 
8.53944 

8-54 308 
Los:. Cot. 



718 

706 
695 

6S4 

673 
663 

653 

643 

634 
625 

616 

607 

599 
591 

583 
575 
568 
560 

553 
546 
539 
533 
527 
520 

514 
50S 
502 

496 
491 
4S5 
480 

475 
469 
464 
460 

455 
450 

445 
441 
437 
432 
428 
424 
419 
416 
412 
408 
404 
400 
396 
393 
390 
386 

383 
379 
376 
373 
370 

366 

364 

Com, l». 



I-oir. Cut. 


Loir. Cos. 


1,75 808 


9-99 993 


1.75090 


9-99 993 


1.74383 


9-99 993 


1.73 688 


9.99992 


1.73004 


9.99992 


1.72 331 


9.99992 


1. 71 667 


9.99992 


1. 71 014 


9.99992 


1.70 371 


9.99991 


1.69736 


9.99991 


1.69 III 


9.99991 


1.68495 


9.99990 


1.67888 


9.99990 


1.67288 


9.99990 


1.66697 


9.99990 


1.66 1 14 


9.99989 


1.65 539 


9-99989 


1,64971 


9.99989 


1. 64410 


9.99989 


1.63857 


9.99988 


1.63 310 


9.99988 


1.62 771 


9.99988 


1,62 238 


9.99987 


1,61 711 


9.99 9S7 


1. 61 191 


9.99987 


1,60676 


9.99986 


1,60 168 


9,99986 


1.59666 


9.99986 


1.59 169 


9.99986 


1.58678 


9.99985 


1.58 193 


9.99985 


1. 57713 


9-99985 


1.57238 


9.99984 


1,56768 


9-99984 


1,56304 


9.99984 


1,55844 


9.99983 


1-55389 


9.99983 


1-54 938 


9.99982 


1-54 493 


9.99982 


1.54052 


9.99982 


1. 53615 


9.9998! 


1-53183 


9.99981 


1-52754 


9.99981 


1.52330 


9.99980 


1. 51 911 


9,99980 
9-99 979 


1. 51 495 


1. 5 1 083 


9-99 979 


1.50675 


9-99 979 


1.50270 


9-99 978 


1.49870 


9.99978 


1-49 473 


9.99978 


1,49080 


9.99977 


1,48690 


9-99 977 


1.48 304 


9-99 976 


1,47921 


9.99976 


1,47541 


9-99 975 


1,47 165 


9-99 973 


1,46792 


9-99 975 


1,46422 


9-99 974 


1,46055 

1.45 691 


9-99 974 


9.99973 


Loir, I'jin. 


Loir. Sin. 



ss 



349 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 









2 













/ 


Log. Sin. 


D 


Log. Tan. 


Com. D. L 


og. Cot. 


Log. Cos. 







8.54 282 


360 


8.543O8 


■^(m 


45691 


9-99 973 


60 


I 


8.54642 


8.54669 




45331 


9 


99 973 


59 


2 


8.54999 


357 


8.55027 


358 J 


44 973 


9 


99972 


58 


3 


8-55 354 


354 


8.55381 


354 J 


44618 


9 


99972 


57 , 


4 


8-55 705 
8.56054 


351 

348 


8-55 733 
8.56083 


352 J 


44266 


9 


99971 


56 


5 


349 J 
346 J 


43917 


9 


99971 


55 


6 


8. 56 400 




8.56429 


43571 


9 


99971 


54 


7 


8.56743 


343 


8.56772 


343 I 


4322^ 


9 


99970 


53 


8 


8.57083 


340 


8.57 113 


341 I 


42886 


9 


99970 


52 


: 9 


8. 57 421 


338 
335 


8.57452 


338 I 


42 548 


9 


99969 


51 


10 


8.57 756 


^•Sll^l 


335 


.42 212 


9 


•99969 


50 


II 


8.58089 


332 


8.58 121 


333 J 


41879 


9 


99968 


49 


12 


8.58419 


330 


8.58451 


330 I 


41 548 


9 


99 968 


48 


13 


8.58747 


327 


8.58779 


328 J 


41 220 


9 


99967 


47 


14 


8.59072 


325 
323 


8.59105 
8.59428 


325 I 


40895 


9 


99967 


46 


15 


8-59 395 


323 J 


40571 


9 


99966 


45 i 


i6 


8.59715 


320 


8.59749 




40251 


9 


99966 


44 i 


'7 


8.60033 


3^3 


8.60067 


318 I 


39932 


9 


99965 


43 


i8 


8.60349 


310 


8.60384 


316 I 


.39616 


9 


99965 


42 1 


19 


8.60662 


313 
311 


8.60698 


314 I 


39302 


9 


99 964 


41 


20 


8.60973 


8.61 009 


6^'- "■ J 


38990 


9 


99964 


40 


1 "^ 


8.61 282 


309 


8.61 319 


309 J 


38681 


9 


99963 


39 


22 


8.61 589 


306 


8.61 626 


307 I 


38374 


9 


99963 


38 


23 


8.61 893 


304 


8.61 931 


3C5 I 


38068 


9 


99962 


37 


1 24 


8.62 196 


302 
300 


8.62 234 


303 I 


37765 


9 


99962 


?>^ 


1 25 


8.62 496 


8.62 535 


300 


37465 


9 


99.961 


35 


i 26 


8.62795 


298 


8.62 834 


299 I 


37 166 


9 


99961 


34 1 


V 


8.63 091 


296 


8-63 131 


297 I 


36869 


9 


99960 


33 


28 


8-6338! 


294 


8.63425 


294 J 


36574 


9 


99 959 


32 


29 


8.63677 


292 
290 


8.63 718 


293 I 


36281 


9 


99 959 


31 1 


1 30 


8.63968 


8.64009 


291 


35990 


9 


99 958 


30 


31 


8.64256 


283 


8.64 298 


283 J 


35702 


9 


99958 


29 


32 


8.64 543 


285 


8.64585 


287 - 


35414 


9 


99 957 


28 1 


33 


8.64827 


284 


8.64870 


285 I 


35 129 


9 


99 957 


27 i 


34 


8.65 no 


282 
281 


8.65153 


283 I 


34846 


9 


99 956 


26 ! 


35 


8.65 391 


8.65435 


2S1 


34565 


9 


99956 


25 


36 


8.65670 


279 


8.65715 


280 


34285 


9 


9995! 


24 


37 


8.65 94f 


277 


8.65 993 


278 J 


34007 


9 


99 954 


23 


38 


8.66223 


275 


8.66 269 


276 J 


33731 


9 


99 954 


22 


1 39 


8.66497 


274 
272 


8.66 543 


274 I 


33 456 


9 


99 953 


21 


1 40 


8.66 769 


8.66816 


-/- 


33184 


9 


99 953 


20 


41 


8.67 039 


26§ 


8.67087 


271 ^ 


32913 


9 


99952 


19 


42 


8.67 308 


8.67 356 




32643 


9 


99952 


18 


43 


8-67575 




8.67624 




32376 


9 


99951 


17 


44 


8.67 840 


265 

264 


8.67 890 


266 . 


32 no 


9 


99950 


16 : 


1 45 


8.68 104 


8.68 154 


262 


31845 


9 


99950 


15 


46 


8.68 366 




8.68417 


31 583 


9 


99 949 


14 


47 


8.68627 




8.68 678 




31 321 


9 


99 948 


13 


48 


8.68 886 


259 


8.68938 


259 J 


31 062 


9 


99948 


12 


1 49 


8.69 144 


257 

256 


8.69 196 


258 I 


30803 


9 


99 947 


II 


50 


8.69400 


8.69453 


256 


30547 


9 


99 947 


10 ! 


51 


8.69654 


254 


8.69708 


255 J 


30 292 


9 


99 946 


9 


52 


8.69907 


253 


8.69 961 


253 J 


30038 


9 


99 945 


8 


53 


8.70159 


251 


8.70214 




29786 


9 


99 945 


7 


54 


8. 70 409 


250 
248 


8.70464 


250 , 


29 53? 


9 


99 944 


6 


55 


8.7065^ 


8.70714 


249 
248 


29286 


9 


99 943 


5 1 


56 


8.70905 


247 


8.70962 


29038 


9 


99 943 


4 ! 


57 


8.71 150 


245 


8.71 208 


246 , 


28 791 


9 


99942 


3 1 


58 


8.71 395 


244 


8.71453 


245 J 


28546 


9 


99942 


2 ; 


57 


8.71 638 


243 
241 


8.71 697 


243 J 


28303 


9 


99941 


I i 


60 


8.71 880 


8.71 939 


242 


28060 


9- 


99940 


^ 1 




Log. Cos. 


D 


Log. Cot. 


Com. D. Lc 


)g. Tan. 


Log. Sin. j 


1 



87' 



350 



TABLE VII. — LOGARITHMIC SINES, COSIXES, TANGENTS, AND C0TAN(;ENTS 



10 

II 

12 

13 
14 



25 
26 
27 
28 

30 

31 

32 
33 

34 

35 
36 
37 
38 
39 
40 

41 
42 

43 
44 

45 
46 

47 
48 

49 

oO 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Log. Sin. 

8.71 880 

8.72 120 
8.72359 
8.72597 
8.72833 



8.73069 
8.73 302 

8-73 533 
^■73 766 
8.73997 



8.74 226 

8-74453 
8.74680 
8.74905 

8.75 129 



8-75 353 
8-75 574 

8-75 795 
8.76015 
8.76233 



8.76451 
8.7666; 
8.76883 
8.77097 
8.77310 



8.77 522 

8.77 733 
8.77943 

8.78 152 
8.78360 



8.78 567 
8.78773 
8.78978 
8.79183 

8.79386 



8.79588 

8.79789 
8.79989 
8.80189 
8.80387 



8.80585 
8.80782 
8.80977 
8.81 172 
8.81 366 



8.81 560 
8.81 752 

8.81 943 

8.82 134 
8.82324 



8.82 513 
8.82 701 
8.82888 
8.83075 
8.83260 



^-83445 
8.83629 
8.83813 

8.83995 
8.8417; 



8-84 358 
Log. Cos. 



240 
239 
237 
236 
235 
233 
233 
231 
230 
229 
227 

226 

225 

224 

223 

221 

221 

219 

2I§ 

217 

215 

215 

214 

213 

212 

211 

210 

209 

208 

207 

206 

205 

204 

203 

202 

201 

200 

199 

198 
197 

197 

195 
195 
194 
193 
192 
191 
191 



187 
186 

185 
185 
184 

183 
182 
182 
181 



d. 



Loy. Tan. 

8-71 939 
8.72186 
8.72420 
8.72 659 
8.72896 



8.73 131 

8-73366 

8-73 599 
8.73831 

8.74062 



8.74 292 

8.74 520 
8-74748 

8.74974 

8.75 199 



8.75422 
8.7564$ 
8.75867 
8.76087 
8.76 306 



8.76524 
8.76741 
8.76958 
8.77 172 

8.77386 



8.77 599 
8.77 811 
8.78022 
8.78232 
8.78441 

8.78648 
8.7885$ 
8.79061 
8.79 266 
8.79470 



8.79673 

8.79875 
8.80075 

8.80275 

8.80476 



8.80674 
8.80871 
8.81 068 
8.81 264 
8.81 459 



8.81 653 

8.81 846 
8.82038 

8.82 230 
8.82 420 



8.82616 

8.82 799 
8.82987 
8.83175 

8.83 361 



8.83547 
8.83732 

8.83 916 

8.84 100 
8.84282 
8.84464 
Log. Cot. 



c. d. 

241 

240 

2J8 

237 
235 
235 
233 
232 
231 
229 

228 
227 
226 
225 
223 
223 
221 
220 
219 
218 
217 
216 
214 
214 

213 
212 
210 
210 

209 
207 
207 
206 
204 
204 
203 
202 
201 
200 
199 
198 
197 
197 
195 

195 
194 

193 
192 
191 
190 
190 



187 
186 
185 
185 
184 
183 
182 
182 



3° 

Lotf. Cot. 
1.28 066 
1.27 819 
1.27 579 
1.27 341 
1.27 104 



1.26868 
1.26633 
1 . 26 400 
1.26 168 
1.25937 



1.25 708 
1.25479 
1.25 252 
1.25 026 
1.24 801 



1.24577 

1.24 35-? 
1-24133 
1-23913 

•3 693 



I. 



1-23475 
1.23258 
1.23 042 
1.22 82; 
1.22 613 



1.22 400 
1.22 188 
1. 21 978 
1. 2 1 768 
1. 21 559 



1. 21 351 
1. 21 144 
1.20 938 
1.20734 
1.20 530 



1.20 327 
1.20 125 
1. 19923 
1. 19 723 
1. 19 524 



1. 19 326 
1. 19 128 
1. 18 931 
1. 18736 
1. 18 541 



1. 18 347 
1. 18 154 
1. 17 961 
1. 17 770 
1. 17 579 



1-17389 
1. 17 201 
1. 17 012 
1. 16825 
1. 16 638 



1. 16453 
1. 16 268 
1. 16083 
1 . 1 5 900 
1.1571; 
^•15535 

c. (1. Log. Tan. 



Loe. Cos. 



9.99940 
9.99940 

9-99 939 
9-99 938 
9.99938 



9.99937 
9-99 936 
9-99 935 
9-99 935 
9-99 93-+ 



9-99 933 
9-99 933 
9-99932 
9-99 931 
9-99931 



9.99930 

9-99929 
9.99928 
9.99928 

9-99927 



9-99926 
9.99925 

9-99925 
9-99924 
9.99923 



9.99922 
9.99922 
9.99921 
9.99926 
9.99919 



9.99919 

9.99918 

9.9991; 

9-99 916 
9.99916 



9.99915 
9.99914 

9-99913 
9.99912 

9.99912 



9.99 911 
9.99910 
9-99 909 
9-99 908 
9.99907 



9.99907 
9.99906 

9-99905 
9.99904 
9.99903 



9.99902 
9.99902 
9.99901 
9.99900 

9-99899 

9.99898" 

9.99897 

9.99895 

9.99896 

9-99895 

9-99894 

Lug. Sin, i 



50 

49 
48 

47 
_46^ 

45 
44 
43 
42 
41 



40 

39 
38 
37 
_3i 
35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 

23 
22 

21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 

9 

8 

7 
6 



r. r. 





330 


320 


310 


6 


330 


32.0 


31 


7 


38.5 


37-3 


36.1 


8 


44.0 


42-6 


41 3 


9 


49-5 


48.0 


46.5 


10 


55 -o 


53-3 


S'O 


20 


110. 


106.6 


J03-3 


30 


165.0 


160.0 


1550 


40 


220.0 


213.3 


206. A 


50 


275.0 


266.6 


258.3 





290 


280 


270 


6 


29.0 


28.0 


27.0 


7 
8 


33-8 
38.6 


32-6 
37-3 


360 


9 

10 


43 S 
48.3 


42.0 
46.6 


40.5 
450 


20 


96.6 


93-3 


90.0 


30 
40 


145.0 
193-3 


140.0 
186.^ 


135-0 
180.0 


5'^ 


241.6 


233 -3 


225.0 



300 

30.0 

350 

40.0 

45.0 

50.0 
100.0 
150.0 
200.0 

250.0 



260 

26.0 

30-3 

34-6 
39 o 
43-3 
86.6 
130.0 

173-3 
216.6 





250 


240 


230 


220 


6 


25.0 


24.0 


23.0 


22.0 


7 


29 


I 


28.0 


26 


§ 


25 


6 


8 


33 


3 


32.0 


30 


6 


29 


3 


9 


37 


5 


36.0 


34 


5 


33 





10 


41 


6 


40.0 


38 


3 


36 


ft 


20 


83 


3 


80.0 


76 


6 


73 


3 


30 


12s 





120.0 


IIS 





no 





40 


166 


6 


160.0 


153 


3 


146 


6 


50 


208 


3 


200.0 


191 


6 


X83 


3 





210 


200 


190 


180 


6 


21 .0 


20.0 


19.0 


18.0 


7 


24-5 


23-3 


22.1 


21 .0 


8 


28.0 


26.6 


25-3 


24.0 


9 


31-5 


30.0 


28.5 


27.0 


10 


35-0 


33-3 


31-6 


30.0 


20 


70.0 


66.6 


63-3 


60.0 


30 


105.0 


100.0 


05.0 


90.0 


40 


140.0 


1.33-3 


126.6 


120.0 


50 


175-0 


166.6 


158-3 


150.0 



6 


9 

0.9 


9 

0.9 


8 

0.8 


7 

0.7 


6 

0.6 


7 

8 


i.i 
1.2 


1.0 
1.2 


0.9 
1.0 


0.8 
0.9 


0.7 
0.8 


9 


1.4 


1-3 


1.2 


1 .0 


0.9 


10 


1.6 


1-5 


1-3 


1.1 


1.0 


20 


31 


3-0 


2-6 


2-3 


2.0 


30 
40 
50 


4-7 

6-3 
7-9 


4 5 
6.0 

7 5 


4.0 
6-6 


3-5 
5-8 


30 
4.0 
5-0 



0.5 
0.6 

o 6 

0.7 



2-5 

3| 
4-1 





4 


4 


3 


2 


I 


6 


0.4 


0.4 


0.5 


0.2 


0.1 


7 


o.,S 


0.4 


0.3 


0.2 


0.1 


8 


0.6 


0.5 


0.4 


0.2 


I 


9 


°? 


0.6 


0.4 


0.3 


0.1 


10 


0.7 


0-^ 


O..S 


0.3 


0.1 


20 


1-5 


'■3 


I.O 


0.6 


03 


30 


2.2 


2.0 


>-5 


1.0 


0-5 


40 


3? 


^•6 


2.0 


'•? 


0$ 


5<J 


3-7 


3-3 


2-5 


1-6 


0-8 



0-3 
0.4 



v. r 



351 



TABLE VII.— LOGARITHMIC SINES/COSINES, TANGENTS, AND COTANGENTS. 

4° 



10 

II 

12 



i6 

17 
i8 

19 



20 

21 



24 



Loir. Niii. 



^•^4 358 
8.84 538 
8.84 718 
8.84897 

8.85075 



8.85 252 
8.85 429 
8.85605 
'8.85780 
8.85954 



8.86128 
8. 86 301 
8.86474 
8.86645 
8.86 816 



"25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 
43 
44 

45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



8.86987 
8.87156 
8.87325 
8.87494 
8.87661 




8.87828 

8.87995 
8.88 160 
8.88326 
8.88490 



8.88654 
8.8881^ 
8.88980 

8.89 142 
8.893 03 
8.89464 
8.89624 
8.89784 

8.89943 

8.90 1 01 



8.90 259 
8.90417 
8.90573 
8.90729 

8.90885 

8.91 040 
8.91 195 
8.91 349 
8.91 502 
8.91655 



8.91 807 

8.91 959 

8.92 no 
8.92 261 
8.92 411 



8.92 561 
8.92 710 
8.92858 
8.93007 

8.93154 



8.93 301 
8.93448 

8.93 594 
8.93 740 
8.93885 



5.94029 



Log. €os. 



d. 



78 
7S 
77 
76 
76 

75 
74 
74 
73 
72 

71 
71 
70 
69 
69 

68 
67 
67 
66 
65 
^5 
64 
63 
63 
62 
62 
61 
61 
60 
59 
59 
58 
58 

57 
56 
56 
56 
55 
54 
54 
53 
53 
52 
51 
51 
50 
50 
50 
49 
48 
48 
47 
47 
46 
46 
46 
45 
44 



Log. Tan. 



8.84464 
8.84645 
8.84826 
8.85005 
8.85 184 



8.85363 
8.85 540 
8.85717 
8.85893 
8.86068 



8.86243 
8.8641^ 
8.86596 
8.86763 
8.86935 



8.87 I06 
8.87277 

8.87447 
8.87616 

8.87785 



8.87953 
8.88 120 

8.88287 
8.88453 
8.88 618 



8.88783 

8.88 94^ 

8.89 III 
8.89274 
8.89436 



8.89598 

8.89759 

8.89 926 

8.90086 

8.90 240 



8-90398 
8.90557 
8.90714: 
8.90872 
8.91 028 



8.91 184 
8.91 340 
8.91 495 
8.91 649 
8.91 803 



8.91 957 

8.92 109 
8.92 262 
8.92413 
8.92 565 



8.92715 
8.92866 
8.93015 
8.93 164 
8-93313 



8.93461 
8.93 609 

893756 
8.93903 
8.94049 



8.94195 



c. d. 



Log. Cot. 



79 
79 
78 
77 
76 
76 

75 

75 
74 
73 

72 
72 

71 

70 
70 
69 



Log. Cot. 



I-I5 535 
1. 15 354 
1. 15 174 
1. 14 994 
1.14815 



1. 14637 
1. 14459 
1. 14283 
1. 14 107 
1-13931 



I-I3756 
1. 13 582 

1.13409 
1. 13237 
1. 13065 



1. 12 893 
1. 12 723 

I.I2 553 
1. 12 384 
1. 12 215 



1 . 1 2 047 
I. II 880 
I. II 713 
1. 1 1 547 
I. II 381 



I. II 216 

I. II 052 

1. 10 889 

1. 10726 

1. 10 563 



1. 10 401 
1. 10 246 
1. 10 079 
1.09 919 
1.09 760 



c. d. 



1.09 601 
1.09443 
1.09285 
1.09 128 
1.08 971 



1.08 815 
1.08660 
1.08 505 
1.08356 
1.08 196 



1.08 043 
1.07 896 
1.07 738 
1.07 586 
1.07435 



1.07 284 
1.07 134 
1.06 984 
1.06 835 
1.06686 



1.06 538 
1.06 396 
1.06 243 
1.06097 

I-05 950 
1.05 805 



Log. Tan. 



85 



Log. Cos. 



9.99894 
9.99893 
9.99892 

9-99891 
9.99896 



9.99 889 
9.99888 
9.99888 
9.99887 
9.99886 



9.99885 
9.99884 
9.99883 
9.99 882 
9.99881 



9.99886 
9.99879 
9.99878 
9.9987^ 

9-99876 



9.99875 
9.99874 
9.99874 

9-99873 
9.99872 



9.99871 
9.99 870 
9.99869 
9.99868 
9-99867 



9.99 866 
9.99 865 
9.99 864 
9.99863 
9.99 862 



9.99 861 
9.99 860 

9-99859 
9-99858 
9.99857 



9.99856 
9-99855 
9-99853 
9.99852 

9-99851 



9.99856 
9.99849 
9.99848 
9-99847 
9-99846 



9.99845 
9.99844 
9.99843 
9.99842 
9.99841 



9.99840 
9.99839 
9-99837 
9-99836 
9-99835 



9.99834 



Log. Sin. 



60 

59 
58 
57 
56 



55 

54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 

26 



25 
24 

23 

22 

21 
"20 

19 
18 

17 
16 



15 

14 

13 
12 

II 

lo 

9 

8 

7 
6 



p. P. 





x8i 


180 


178 


176 


6 


18. 1 


18.0 


17.8 


17.6 


7 


21. 1 


21.0 


20 


7 


20.5 


8 


24.1 


24.0 


23 


7 


234 


9 


27.1 


27.0 


26 


7 


26. 4 


10 


30.1 


30.0 


29 


6 


29-3 


20 


60.3 


60.0 


59 


3 


58.6 : 


30 


90-5 


90.0 


89 





88.0 


40 


120.6 


120.0 


118 


6 


117-3 


50 


150-8 


150.0 


148 


3 


146.6 1 





174 


172 


170 


6 


17.4 


17.2 


17.0 


7 


20.3 


20.0 


19-8 


b 


23. 2 


22.9 


22-6 


9 


26.1 


25. « 


25. s 


10 


29.0 


28.6 


28.3 


20 


58.0 


57-3 


.56.6 


30 


87.0 


86.0 


85.0 


40 


116. 


114. 6 


"3-3 


50 


145-0 


143-3 


141. 6 



6 


166 

16.6 


164 

16.4 


162 

16.2 


7 
8 


19-3 
22.1 


19.1 

21-8 


18.9 
21.6 


9 


24.9 


24.6 


24-3 


10 


27-6 


27-3 


27.0 


20 
30 


55-3 
83.0 


54-6 
82.0 


54-0 
81.0 


40 
50 


no. 6 

138.3 


109.3 

136.6 


108 .0 

135-0 





158 


i.S6 


I.S4 


6 


15-8 


15.6 


15-4 


7 


18.4 


18.2 


17.9 


8 


21.0 


20.8 


20.5 


9 


23-7 


23-4 


23-1 


10 


26.3 


20. 


25-6 


20 


52.6 


52.0 


51-3 


30 


79.0 


78.0 


77.0 


40 


105 -3 


104.0 


102.6 


50 


131-6 


T30.0 


128.3 



168 

16.8 

19.6 
22.4 
25.2 

28.0 

ce.o 

84.0 

112.0 

140.0 



160 

16.0 
18.6 
21.3 
24.0 

26 6 

53-3 

80.0 

106.6 

133-3 



15.2 
17.7 
20.2 
22.8 
25-3 
50.6 
76.0 
101.3 
126.6 





146 


145 


1^ 

I 


I 


6 


T4.6 


14-5 


0.1 


0. 1 


7 


17.0 


16 


Q 


0.2 


0. 1 


8 


19 4 


19 


3 


2 


0.1 


9 


21.9 


21 


7 


0.2 


0.1 


10 


24-3 


24 


I 


0.2 


0.1 


20 


48.6 


48 


3 


05 


0-3 


30 


73.0 


72 


5 


0.7 


o-s 


40 
50 


97-3 
121. 6 


96 
120 


6 
8 


x.o 
1.2 


°-6 

0-8 



P. p 





150 


149 


148 


147 1 


6 


15.0 


14.9 


14.8 


14.7 


7 


17-5 


17 


4 


17 


2 


17.1 


8 


20.0 


19 


8 


19 


7 


19.6 


9 


22.5 


22 


3 


22 


3 


22.0 


10 


25.0 


24 


8 


24 


6 


24-5 


20 


50.0 


49 


6 


49 


3 


49.0 


30 


75-0 


74 


5 


74 





73-5 


40 


100.0 


99 


3 


98 


6 


98.0 


50 


125.0 


124 


1 


123 


3 


122.5 



0.0 
0.0 



352 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGEN IS, AND C()TAN(iENTS. 



;> 



10 

II 

12 
13 

15 
16 

17 
18 

19 

20 

21 

22 

23 
24 



25 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



Lot?. Sip. I d. 



50 

51 

52 
53 

54 



55 
56 
57 
58 

00 



8.94029 
8.94174 
8.94317 
8.94460 
8.94603 



8.94745 
8.94887 

8.95028 
8.95 169 
8.95310 



8.95450 

8.95 589 

8.95728 
8.95867 

8.96 005 



8.96 143 
8.96280 
8.96417 

8-96553 
8.96689 



8.96825 
8.96960 
8.9709^ 
8.97 229 
8.97 363 



8.97496 
8.97 629 
8.97 762 
8.97894 
8.98026 



8.Q8 ii;7 
8.98288 
8.98419 
8.98 549 
8.98679 



8.98 808 
8.98937 
8.99066 
8.99194 
8.99322 



8.99449 
8.99 577 
8.99703 
8.99830 
8.99956 



9.00081 
9.00 207 
9.00 332 
9.00456 
9.00 580 



9.00704 
9.00828 
9.00951 
9.01 073 
9.01 196 



9.01 318 
9.01 440 
9.01 561 
9.01 682 
9.01 803 

9.QI 923 

Log. Cos. 



144 
143 

143 
143 
142 
142 
141 
141 
140 
140 
139 
139 
138 
138 
138 
137 
137 

136 
136 
135 
135 
134 
»34 
134 
133 
133 
132 

132 
132 
131 
131 
130 

130 
130 
129 
129 

12§ 
I2g 

127 
127 
127 
126 
126 

126 

125 
125 
125 

124 
124 

124 

123 

123 

122 
122 
122 
122 
121 
121 
120 
126 



Log. Tan. | c. d. | Log. <'<>t. 



8.94 195 
8.94346 
8.94485 
8.94629 
8.94773 



8.94917 
8.95059 
8.95 202 

8.95 344 
8.95485 



8.95626 
8.95767 

8.95 90^ 
8.9604^ 

8.96 i8g 



8.96325 
8.96464 

8.96602 

8.96739 

8.96876 



8.97 013 

8.97 149 
8.97 285 
8.97421 
8.97 556 



8.97 690 
8.97 825 

8.97 958 

8.98 092 
8.98225 



8.98357 
8.98490 
8.98621 

8.98753 
8.98884 



8.9901$ 
8.99 145 



8. 


99275 


8. 


99404 


8. 


99 533 


8. 


99662 


8. 


99791 


8. 


99919 


9 


00046 


9 


00 174 


9 


00 306 


9 


00427 


9 


00553 


9 


00679 


9 


00804 


9 


00 930 



9.01 054 
9.01 179 

9.01 303 

9.01 427 



9.01 550 
9.01 673 

9.01 796 

9.01 9I8 

9. 02 046 
9.02 162 

Log. Cot. 



145 
144 
144 
144 

M3 
142 
142 
142 
141 
141 
141 
140 
140 
139 
139 
138 
138 

137 
137 
137 
136 
136 
135 
135 
134 
134 
133 
133 
133 
132 
132 
131 
131 
131 
131 
130 
130 
129 
129 
129 

I2g 
128 
127 
127 
126 
126 
126 

125 
125 
125 
124 
124 
124 
124 
123 
723 

123 

122 
122 
121 

"cTJT 



1.05 803 
1.05659 
1.05 515 
1.05 370 
1.05226 



1.05 083 
1.04946 
1.04 798 
1.04 656 
1.04 514 



1.04373 

1.04 232 

1.04092 

1.03952 

j_^03_8_i^ 

1.03674 

1.03 536 
1.03398 
1.03 266 
1.03 123 



1.02 986 
1.02 856 
1.02 714 
1.02 579 
1.02 444 



1.02 309 
1.02 175 
1.02 041 
1. 01 908 
1. 01 775 



Lour. Cos. 



1. 01 642 
1. 01 510 
1. 01 378 
1. 01 247 
1. 01 116 



1.00985 
1.00855 
1. 00 725 
1.00595 
1.00466 



1.00337 
1. 00 209 
1 . 00 08 1 

0.99953 
0.99 826 



0.99699 

0.99 573 
0.99 446 
0.99321 

0.99195 



0.99070 

0.98945 
0.98821 

0.98 697 

0.98 573 



0.98 450 
0.98327 
0.98 204 
0.98 081 

0.97959 
0.97 838 

JiOg. Tan. 

S4t' 



9.99834 

9-99833 
9.99832 

9.99831 

9.99830 



9.99829 
9.99827 
9.99826 
9-99825 
9.99824 



9.99823 
9.99 822 
9.99 821 
9.99819 

9.99 81 8 



9.99817 

9-99816 
9.99815 
9.99814 
9-99 Si 3 



9.99 81 1 
9.99 816 
9.99809 
9.99 808 
9.99807 



9.99805 
9-99804 
9.99803 
9.99 802 
9.99801 



9.99799 

9-99 798 
9.99797 

9.99796 
9-99 794 



9-99 793 
9.99 792 
9.99791 
9-99789 
9-99788 



9.99787 
9.99786 
9.99784 

9-99783 
9-99782 



9.99781 
9.99779 

9-99 778 
9.99777 
9.99776 



9-99 774 

9-99 773 
9.99772 

9.99776 

9.99769 



9.99768 

9.99766 

9.99765 
9.99764 

9.99763 



9-99761 

Log. Sin. 



00 

59 
58 
57 
Ji 
55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 

35 
34 
33 
32 
31 



I'. I'. 



30 

29 

28 

27 
26 

25 
24 

23 
22 
21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
1 1 

To 

9 

8 

7 
6 



140 139 138 137 



14.0 


13.9 


13.8 


13-7 


16. s 


16.2 


16. 1 


16.0, 


18.6 


18.5 


18.4 


18.2, 


21 .0 


20.8 


20.7 


20.5 


23-3 


23.1 


23.0 


22.8 


46.6 


46.3 


46.0 


45-6 


70.0 


69-5 


69.0 


68.5 


93-3 


92.$ 


92.0 


9'-3 


116. 6 


i'5-8 


115. 


114.1 



'3-5 


13-4 


13 


3 


»5 7 


15-6 


15 


5 


18.0 


17-8 


17 


7 


20.2 


20. 1 


19 


9 


22.5 


22.3 


22 


1 


45.0 

675 
90.0 


44-6 
67.0 
89.3 


44 
66 
88 


3 

§ 
<5 


112.5 


III. 6 


110 


8 





145 


144 


143 


142 


141 


6 


14-5 


14.4 


14.3 


14.2 


14 I 


7 


16.9 


16.8 


16.7 


16 .S 


16.4 


8 


19-3 


19.2 


19.0 


18.9 


18.8 


9 


21.7 


21.6 


21.4 


21.3 


31. I 


10 


24.1 


24.0 


23 § 


23-6 


23-5 


20 


48-3 


48.0 


47-6 


47-3 


47.0 


30 


72 5 


72.0 


71 5 


71.0 


70.5 


40 


9<> 6 


96.0 


95-3 


94-^ 


94.0 


50 


120.8 


120.0 


119. 1 


118.3 


H7-5 



136 

13.6 

o; 15-8 

- 18. 1 

20.4 

22.^ 

45-3 
68.0 

90-6 

l"3-3 



135 134 133 132 

13.2 

15-4 
17.6 
19.8 
22.0 
44.0 
66.0 





131 


130 


129 


128 


6 


13-1 


13.0 


12.9 


12.8 


7 


153 


151 


15.0 


14 9 


8 


17-4 


173 


17.2 


17.0 


9 


19-$ 


19.5 


193 


19.2 


10 


21. § 


21.6 


2t-5 


21-3 


20 


43-6 


43-3 


43 


42.6 


30 


6s.,S 


65.0 


64.5 


64.0 


40 


87.3 


86.^ 


86.0 


85.3 


50 


109.1 


108.3 


107-5 


106.6 



127 


126 


125 


124 


123 


12.7 


12.6 


12.5 


12.4 


12.3 


14.8 


14-7 


14.6 


14.4 


14-3 


16.9 


16.8 


16.6 


16.5 


16.4 


19.0 


18.9 


18 7 


18.5 


18.4 


21.1 


21.0 


20. § 


20.^ 


20.5 


42.3 


42.0 


41 6 


41-3 


41.0 


63. s 


63.0 


62.5 


62.0 


6t.5 


84-6 


84.0 


83.3 


82. 6 


82.0 


105-8 


105.0 


104.1 


103.3 


102.5 



122 

12.2 
14.2 
16.2 
.8.3 
20.3 
40.6 
61 .0 
81.3 

lOI.fi 



121 

12. 1 
14.1 
i6.i 
18.1 
20. T 
40.3 
6a. s 
80.^ 
100. 8 



120 I 

o.i 
0.2 
0.2 



12.0 

•4 

16.0 
18.0 
20 
40 
60 
80 
100. o 



O. 2 



0-5 

0.7 



I O 



0.0 
0.0 
0.1 
0.1 
0.1 
o.i 
o 3 
0.4 



0-3 
o-S 
o $ 
0-8 



P. P. 



353 



TABLE VII.— LOGARITHMIC 



SINES, COSINES, TANGENTS, AND COTANGENTS. 



10 

II 

12 
14 



15 
16 

18 
19 



20 

21 

22 

23 

24 



^5 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 

39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 

54 



55 
56 
57 
58 
59 



60 



Lo^. Sin. 



9.01 923 

9.02 043 
9,02 163 
9.02 282 
9.02 40T 



9.02 520 
9.02638 

9.02 756 
9.02 874 
9.02 992 



9.03 109 

9.03 22^ 

9-03 342 
9.03458 

9-03 574 



9.03689 

9.03 805 
9.03919 
9.04034 

9.04 148 



9.04 262 
9.04376 
9.04489 
9.04 602 
9.04715 



9.04 828 

9.04 940 

9.05 052 
9.05 163 
9.05275 



9.05 386 

905496 
9.05 607 
9.05717 
9.05 827 



9-05 936 
9.06046 
9.06155 
9.06 264 
9.06 372 



9.06480 
9.06 588 
9.06 696 
9.06 803 
9.06 910 



9.07 017 
9.07 124 
9.07 230 

9-07 336 
9.07 442 



9.07 548 
9.07653 

9-07 758 
9.07 863 
9.07 96^ 



9.08 072 
9.08 176 
9.08 279 
9.08 383 
9.08485 



9.08 589 



Log. Cos. 



(1. 



120 
119 
119 
119 
119 
118 
118 
118 
117 
117 

"6 
"6 
116 
116 
115 
115 
114 
114 
114 
114 

"3 
113 

113 
113 
112 
112 



III 
III 



no 
log 
109 
109 
109 
io§ 
108 
108 
107 
107 
107 
107 
log 
log 
106 
106 

105 
105 

105 

104 
104 
104 
104 
103 
103 
103 
103 



(1. 



Lost. Tan. 



9.02 162 
9.02 283 
9.02 404 
9.02 525^ 
9.02 645' 



9.02 765 
9.02885 

9.03 004 
9.03 123 
9.03 242 



9.03 361 

9-03 479 
9-03 597 
9.03714 
9.03 831 



9-03 948 
9.04065 

9.04 18T 

9.04 297 

9.04413 



9.04 528 
9.04643 
9.04758 
9.04 872 
9.04987 



9.05 lOI 
9.05 214 
9.05 32^ 
9.05 446 
9-05 553 



c. d. 



9.05 666 
9.05778 

9.05 890 
9.06001 

9.06 113 



9.06 224 
9-o6 335 
9.06445 
9.06555 
9.06 665 



9.06775 
9.06 884 

9.06 994 

9.07 102 
9.07 211 



9.07319 
9.07 428 

9-07 53^ 
9.07 643 
9.07 756 



9.07857 

9.07 964 

9.08 071 
9.08 177 
9.08 283 



9.08 389 
9.08 494 
9.08 600 
9.08 705 
9.08 810 



9.08 914 



121 
121 
120 
120 
120 
119 
119 
119 
119 

"8 
118 
118 
117 

117 
117 

"6 
116 



115 
114 
114 
114 
114 
113 
"3 
"3 

"3 
112 

112 
112 
III 
III 
III 
III 
no 
no 
no 
109 
109 
109 
log 
109 
108 
log 
107 
107 
107 
107 
107 
log 
log 
106 

105 
105 
105 
105 
105 

104 



Log. Cot. 



0.97 838 
0.97 7I6 
0.97595 
0.97475 

0.97 354 



0.97234 
0.97 115 
0.96995 
0.96876 
0.96757 



0.96639 

0.96 521 

0.96403 

0.96 285 
0.96 168 



0.96 051 

0-95 935 
0.95 818 
0.95 702 
0.95 587 



0.95471 

0.95 356 
0.95 242 
0.95 127 
0.95013 



0.94899 

0.94785 
0,94672 
0.94559 

0.94 446 



0.94 334 
0.94 222 
0.94 no 

0-93 998 
0.93887 



Lost. Cos. 



9.99761 
9.99760 

9-99 759 
9-99 757 
9.99756 



99 754 

99 753 
99752 
99756 

99 749 



9.99748 
9-99 746 
9-99 745 
9-99 744 
9.99742 



9.99741 

9-99 739 
9-99 738 
9-99 737 
9-99 735 



9-99 734 
9.99732 

9-99731 
9-99730 
9-99 728 



9.99727 

9.99725 
9.99724 
9.99723 
9.99721 



0.93776 
0.93 665 
0.93 554 
0.93444 
0.93334 



0.93 225 
0.93 115 
0.93 006 
0.92 897 
0.92788 



0.92 686 
0.92 572 
0.92 464 
0.92357 
0.92 249 



0.92 142 
0.92035 
0.91 929 
0.91 822 
0.91 716 



0.91 611 
0.91 505 
0.91 400 
0.91 295 
0.91 190 



0.91 085 



Log. Cot*- I c. (1. i Log. Tan. 



83' 



9.99720 

9-99 718 
9.99717 
9-99715 
9.99714 



9.99712 

9-99 71 1 
9.99710 

9-99 708 
9-99707 



9.99705 
9.99 704 
9-99702 
9.99701 
9.99699 



9.99698 
9.99696 
9.99695 

9-99693 
9.99692 



9.99696 
9.99689 
9.99687 
9.99 686 
9.99684 



9.99683 
9.99 681 
9.99679 
9.99678 
999676 



9-99675 



Log. Sin. 



00 

59 
58 
57 
Ji 
55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 

43 
42 

41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
■^i 



30 

29 
28 

27 
26 



25 
24 

23 

22 

21 
"20 

19 
18 

17 
16 



15 
14 

13 
12 

II 



10 

9 
8 

7 
6 



p. p. 



121 121 120 119 118 



6 


12. 1 


12.1 


12.0 


II. 9 


7 
8 


14.2 
16.2 


14.1 

16. i 


14.0 
16.0 


139 
15-8 


9 


18.2 


18. i 


18.0 


17-8 


10 


20.2 


20. 1 


20.0 


19-8 


20 
30 
40 


40-5 
60.7 
81.0 


40-3 

60.5 

80. g 


40.0 
60.0 

80.0 


39-6 
59-5 
79-3 


50 


101.2 


100. § 


100. 


99.1 





II 


7 


117 


116 


6 


n 7 


II. 7 


11.6 


7 


13 


7 


13-6 


13-5 


8 


15 


6 


15-6 


15-4 


9 


17 


6 


J7-5 


17.4 


10 


19 


6 


19-5 


19-3 


20 


39 


I 


39 -o 


.38.6 


30 


5« 


7 


5«-.5 


58.0 


40 


7» 


3 


78.0 


77-3 


50 


97 


9 


97-5 


96. g 



13-7 

15-7 
17.7 

19-6 
39-3 
590 
78. g 



"5 

"•5 



134 

15-3 
17.2 
19. 1 
38.3 

57-5 
76-6 
95-8 





114 


114 


113 


112 


II 


6 


11.4 


II. 4 


"•3 


II. 2 


II. 


7 


13-3 


13-3 


13.2 


13.0 


12. 


b 


15.2 


15.2 


15.0 


14.9 


14. 


9 


17.2 


17. 1 


16.9 


16.8 


16. 


10 


19. 1 


19.0 


18. H 


18. g 


18 


20 


38.1 


38.0 


37-6 


37-3 


37 


30 


57-2 


57-0 


st>-s 


56.0 


55 


40 


76-3 


76.0 


75-3 


74-6 


74 


50 


95-4 


95 -o 


94-1 


93-3 


92 





IIO 


IIO 


109 


6 


n.o 


n.o 


10.9 1 


7 


12.9 


12.8 


12 


7 


8 


14.7 


14 6 


14 


5 


9 


16.6 


16. s 


16 


3 


10 


18.4 


18.3 


18 


I 


20 


36.8 


36-6 


36 


3 


30 


55-2 


55-0 


54 


5 


40 


73-6 


73-3 


72 6 1 


50 


92.1 


91-6 


90 


8 1 





10^ 


107 


106 


105 


6 


10.7 


10.7 


10.6 


10.5 


7 


12. 5 


12.5 


12.3 


12.2 


8 


14-3 


14.2 


14. 1 


14.0 


9 


16. 1 


16 


15-9 


15-7 


10 


17.9 


17-8 


17 6 


17-5 


20 


35-8 


35-6 


35-3 


35-0 


30 


53-7 


53-5 


53-0 


52.5 


40 


71-6 


71-3 


70-6 


70.0 


50 


89.6 


89.1 


88.3 


87-5 





103 


103 


2 


I 


6 


10.3 


10.3 


0.2 


0.1 


7 


12 


I 


12 





0.2 


0.2 


8 


13 


8 


13 


7 


0.2 


0.2 


9 


15 


5 


15 


4 


0.3 


0.2 


10 


17 


2 


17 


I 


0-3 


0.2 


20 


34 


5 


34 


3 


0.6 


0.5 


30 
40 

50 


51 
69 
86 


7 

2 


51 
68 

85 


5 
6 
8 


I.O 

1-3 
1-6 


0.7 
I 
1.2 



108 

10.8 

12.6 

14.4 

16.2 

18.0 

36.0 

540 

72.0 
90 o 



104 

10.4 

12. I 

13-8 

15-6 
17-3 
34-g 
52.0 

69.3 
86-6 



I 

0.1 
0.1 
o.i 
o. I 
0.1 
0-3 
0-5 
o.^ 



P. P. 



354 



TABLE VII. -LOGARITHMIC SINES. COSINES, TANGENTS, AND COTANGENTS. 



10 

II 

12 
14 



i5 
16 

18 
19 



20 

21 



24 



26 
27 
28 
^9 
30 
31 
32 
33 
34 



36 

37 
38 
39 



40 

41 
42 
43 

44 



45 
46 

47 
48 
49 



50 

51 
52 
53 
54 



55 

56 
57 
58 
59 



GO 



Log. Sin. I d. 



9.08 589 
9.08 692 
9.08 794 
9.08 897 
9.08 999 



9.09 lOI 
9.09 202 

9.09 303 
9.09404 
909 50! 

9.09 606 

9.09 706 
9.09 806 

9 09 906 
9 10006 



9. 10 lO^ 
9. 10 205 
9. 10 303 
9. 10 402 
9. 10 501 



9.10599 
9. 10 697 
9.10795 
9. 10 892 
9. 10 990 



91 

9-1 

91 
9.1 

91 



9.1 
9.1 

91 
91 
9-1 



087 
184 
281 
37f 
473 



570 
665 
761 

856 
952 



9. 1 2 047 
9. 12 14T 
9. 12 236 
9.12 330 
9.12425 



9 12 518 
9. 12 612 
9. 1 2 706 
9.12 799 
9. 12 892 



12985 
13078 
13 175 
13 263 
13 355 



9- 1 3 447 

9 13 538 
9.13636 
9.13 721 
9-13 813 
9- 1 3 903 
9- 1 3 994 
9.14085 

9- 14 17! 
9 14265 



9 14355 

Log. Cos. 



99 

99 

99 

98 

99 

98 
98 

98 
97 
97 
97 
97 
96 
97 
96 
96 
96 
95 
96 

95 
9S 
95 
94 
94 
94 
94 
93 
94 
93 
93 
93 
93 
92 
92 
92 
92 
92 

91 
92 

91 
91 
90 

91 
90 
90 
90 
90 

"dT 



Log. Tail. I c. d. 



9.08 914 

9.090I8 

9.09 123 
9.09 226 

9- 09 330 



9-09 433 

909536 
9.09 639 

9.09 742 

9.09 844 



9.09 947 
9.10048 

9.10 150 
9,10 252 
9 10353 



9.10454 
9- 10 555 
9. JO 655 
9.10756 
9.10 856 



9. 10 956 

9. 1 1 055 
9.11 155 
9 II 254 
9ir 353 



9. 1 1 452 
9. 1 1 553 
9. 1 1 649 

9. 1 1 747 
9. II 845 



11 943 

12 040 
12 137 
12 235 
12 331 



9.12428 
9.12 525 
9. 12 621 
9. 12 717 

9.12 813 



9. 1 2 908 

9. 1 3 004 
9. 1 3 099 

9-13 194 
9.13289 



9-13384 
9-13 478 
9.13 572 

9.13665 
9. 1 3 766 



9- 13 854 
9- 1 3 947 
9. 14 041 

9-14134 
9. 14 227 



9.14319 
9.14412 
9.14504 

9-14 596 
9-14688 

9. 14 786 

Lost. Cot. 



104 
104 
103 
103 
103 
103 

103 
102 
102 
102 

loi 
102 

lOI 

101 

lOI 
lOI 

100 
100 
100 

lOO 

99 
99 
99 
99 
98 



98 
98 

97 
97 
97 
96 
97 
96 
96 
96 
96 
95 
95 
95 
95 
95 
94 
94 
94 
94 
94 
93 
93 
93 
93 
93 
92 
92 
92 
92 
92 
92 

c. d. 



Lot'. Cot. 



0.91 085 
0.90 98! 
0.90 877 
0.90773 
0.90 670 



0.90 566 
0.90463 
0.90 360 
0.90 258 
0.90 155 



0.90053 
0.89 95T 
0.89 849 
0.89 748 
0.89 647 



0.89 546 
0.89445 
0.89344 
0.89 244 
0.89 144 



0.89 044 
0.88 944 
0.88845 
0.88745 
0.8 8646 



0.88 548 
0.88 449 
0.88 351 
0.88 253 
0.88 155 



0.88057 
0.87 959 
0.87 862 
0.87 765 

0.87 668 



0.87 571 
0.87475 
0.87 379 
0.87 283 
0.87 187 



0.87 091 
0.86 996 
o. 86 906 
0.86805 
0.86 716 



0.86616 
0.86 521 
0.86 427 
0.86333 
0.86 239 



0.86 146 
0.86 052 
0.85 959 
0.85 866 
0.85773 



0.85 686 
0.85 588 
0.85495 
0.85403 
0.85 31T 



) 85 219 
\AMi. r.-iii. 



\Mii. C(»S. 



9.99675 

9-99673 
9.99672 
9.99676 
9.99669 



9.99667 
9.99665 
9.99664 

9.99 662 
9.99661 

9-99~65§^ 
9.99658 
9.99656 
9.99654 

9-99653 



9-99651 
9.99650 
9.99648 

999646 
9-99645 



9-99643 
9-99641 
9.99640 

9-99638 
9-99637 



9-99635 
9-99633 
9.99632 

9.99630 
9.99628 



9.99627 
9.99625 
9.99623 
9.99622 
9.99 620 
9.99"6Tf 
9.99617 

9-99615 
9.99613 
9.99 6 iT 



9.99 610 
9-99608 
9.99606 
9.99605 
9-99603 



9.99 60T 
9.99 600 
9-99598 
9-99 596 
9-99 594 



9-99 593 
9-99 591 
9-99 589 
9-99 587 
9-99 586 



9.99 584 
9.99582 
9.99586 

9-99 579 
9 99 577 



9 99 5/5 

Loir. Sin. 



(>0 

59 

58 

57 

_56 

55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 

43 
42 

41 



40 

39 
38 
37 
36 



35 

34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 

23 

22 

21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 

To 

9 

8 

7 
6 



20 

30 
40 
50 



I'. I'. 



104 103 102 lOI 



10.4 


10.3 


10.2 


10. 1 


12 I 


12.0 


11.9 


11.8 


13-8 


^3-7 


13-6 


13-4" 


15 6 


I5-4 


'5 3 


'5-1 


173 


17.1 


17.0 


16 I 


34-6 


34-3 


34 


33-6 


52.0 
693 
86.6 


5'-5 
68.6 

85-8 


510 
68.0 
85.0 


50- 5 
^7-3 
84.! 





100 


100 


99 


6 


10. 


10. 


9-9 i 


7 


II. 7 


II. g 


"■5 


8 


13-4 


'3-3 


13.2 


9 
10 


151 
16.7 


15.0 
16.6 


14-8 
16.5 


20 


33-5 


33-3 


33-0 


30 
40 

50 


50.2 
67.0 
83-7 


50.0 
66.6 
83.3 


49-5 
66.0 

82.5 



98 

9.8 

II. 4 
13.0 
14.7 
16.3 

32-6 
49.0 

65.3 
Si. 6 





97 


97 


96 


95 


6 


9-7 


9-7 


9.6 


9-5 


7 


II. 4 


II. 3 


II. 2 


11 1 


8 


130 


12.9 


12.8 


12.6 


9 


14.6 


145 


14.4 


14 


2 


10 


16.2 


16. i 


16.0 


15 


§ 


20 


325 


32 -3 


32.0 


31 


6 


30 


48.7 


48.5 


48.0 


47 


5 


40 


65.0 


64.6 


64.0 


63 


3 


50 


81.2 


80.8 


80.0 


79 


I 





91 


91 


90 


2 


6 


9.1 


9.1 


9.0 


0.2 


7 


10.7 


10.6 


10.5 


0.2 


U 


12.2 


12. 1 


12.0 


0.2 


9 


13-7 


i3-§ 


'3-5 


0.3 


10 


15.2 


15.1 


I5-0 


° 3 


20 


30.5 


30. 3 


30.0 


0.6 


30 


45-7 


45-5 


45.0 


I.O 


40 
50 


61 .0 
76.2 


60.^ 
75-8 


60.0 
75 


'■2 

»-6 



0.5 

0.7 





94 


94 


93 


92 


6 


9.4 


9-4 


9-3 


9.2 


7 


II. 


10.9 


10 


R 


10.7 


8 


12.6 


12-3 


12 


4 


12.2 


9 


14.2 


14.1 


13 


9 


13-8 


10 


'5-7 


i5-$ 


15 


5 


'5-3 


20 


3'-5 


31-3 


3> 





30 6 


30 


47 2 


47.0 


46 


S 


46.0 


40 


63.0 


62. A 


62 





61. 1 


50 


78.7 


78-3 


77 


5 


76.1 



r. I' 



8*e° 



355 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. AND COTANGENTS. 

8° 



Log. hill d. Log. Tan. 



10 

II 

12 

13 

14 



15 
16 

18 
19 



20 

21 

22 

23 

24 



25 
26 

27 
28 

29 



30 

31 
32 
33 

34 



35 
36 

37 
38 
39 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 

54 



55 
56 
57 
58 
59 



60 



9-H355 
9.14445 

9-14 535 
9. 14 624 

9-14713 



9. 14 802 
9.14891 
9.14 980 
9.15068 
915 157 



9.15245 

9-15 333 
9. 15 421 

9-15 508 
9-15595 



9.15683 
9.15770 
9.15857 

9-15 943 
9. 16 030 



9.16 116 
9.16 202 
9.16 283 
9.16374 
9. 1 6 460 



9.16545 
9.16 630 
9.16 716 
9.16 801 
9.16885 



9.16 970 
9.17054 

9.17 139 
9.17 223 
9.17307 



9.17 391 
9 17474 
9.17558 
9-17641 
9.17724 



9.17 807 

9.17 890 
9.17972 
9.18055 

9.18 137 



9. 18 219 
9.18 301 
9.18383 
9.18465 
9.18 546 



9.18628 
9. 1 8 709 
9. 1 8 790 
9.18 871 
9.18 952 



9. 19032 
9.19113 
9.19 193 

9 19273 
9-19353 



9-19433 



Log. Cos. 



90 
89 
89 
89 
89 
89 
88 



88 
88 
88 
87 
S? 
87 
87 
87 
86 
86 

86 
86 
86 
86 
85 
85 
85 

85 

85 
84 

84 
84 
84 
84 
84 
84 
83 
83 
83 
83 
83 
83 

82 
82 
82 
82 
82 
82 
81 
81 

81 
81 
81 
80 
81 
80 

86 
80 
80 
80 

79 



9.14 780 
9.14872 
9.14963 
9.15054 

9.15 145 



9-15236 
9.15327 

9-15417 
9.15 507 
9.15 598 



9.15687 

9-1577? 
9.15867 

9-15 956 
9.16045 



c. d. 



16 134 
16 223 
16 312 
16 401 
16 489 



9 16577 
9.16665 
9.16753 
9. 16 841 
9.16928 



17015 
17 103 
17 190 

17276 
17363 



9- 
9- 
9- 
9- 
9: 

9.17450 
9.17 536 

9.17 622 

9.17708 

9 17794 



9 17880 
9.17965 
9. 18 051 
9.18 136 
9. 18 221 



9. 1 8 306 
9.18 390 

9-18475 
9.18559 

9. 1 8 644 



9- 


18728 


9. 


18812 


9- 


18896 


9- 


18979 


9- 


19063 


9- 


19 146 


9- 


19 229 


9- 


19312 


9- 


19395 


9- 


19478 


9- 


19 566 


9- 


19643 


9- 


19725 


9- 


1980^ 


9- 


19889 



9. 19 971 



91 
91 
91 
91 

91 
96 
96 
90 
90 

89 

90 

89 

89 
89 

89 

89 

89 

88 



87 
88 

87 

S7 

87 
87 
86 
87 

86 
86 

86 
86 

85 
86 



85 
85 
85 
84 
84 
84 
84 
84 

84 
84 
83 
83 

83 
83 
83 
83 
82 

82 
82 
82 
82 

82 
82 



Log . Cot. I c. d. 



tiog. Cot. 



0.85 219 
0.85 128 
0.85037 
0.84945 
0.84854 



0.84763 
0.84673 
0.84582 
0.84492 
0.84 402 



0.84 312 
0.84 222 
0.84 133 

o. 84 043 

0-83954 
0.83865 
0.83776 
0.83687 

0-83 599 
0.83 511 



0.83 422 
0.83334 
0.83 247 
0.83 159 
0.83 071 



0.82 984 
0.82 897 
0.82 810 
0.82 723 
0.82636 



0.82 550 
0.82 464 
0.82 377 
0.82 291 
0.82 206 



0.82 120 
0.82 034 
0.81 949 
0.81 864 
0.81 779 



0.81 694 
0.81 609 
0.81 525 
0.81 446 
0.81 356 



0.81 272 
0.81 188 
0.81 104 
0.81 026 
0.80937 



0.80 854 
0.80 770 
0.80687 
o. 80 604 
0.80 522 



0.80439 
0.80357 
0.80 274 
0.80 192 
0.80 II 6 



0.80 023 



JiOg. Tan. 



81 



Log. Cos. 



9-99 575 
9-99 ^7% 
9.99 371 
9.99 570 
9-99 568 



9.99 566 
9-99564 

9-99563 
9.99561 

9-99 559 



9-99 55? 
9-99 555 
9-99 553 
9.99552 
9.99550 



9-99 548 
9-99 546 
9-99 544 
9-99 542 
9.99541 



9-99 539 
9-99 537 
9-99 535 
9-99 533 
9-99 531 



9-99 529 
9.99528 
9.99526 

9-99 524 
9.99522 



9.99520 
9-99 518 
9-99 516 
9.99514 
9.99512 



9-99 511 
9-99 509 
9-99 507 
9-99 505 
9-99 503 



9-99 501 
9.99499 

9-99 497 
9.99495 
9-99 493 



9.99491 
9.99489 
9.99487 
9.99485 
9.99484 



9.99482 
9-99480 
9.99478 
9-99476 
9-99 474 



9.99472 
9.99470 
9-99468 
9.99466 
9.99464 



9.9946: 



Log. Sin. 



60 
59 

58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



25 
24 

23 
22 

21 



20 

19 
18 

17 
16 



15 

14 

13 
12 

II 

To" 
9 







p. p. 





91 


91 


90 


89 


6 


9.1 


9-1 


9-0 


8.9 


7 


10.7 


10.6 


10.5 


10.4 


8 


12.2 


12.T 


12.0 


II. 8 


9 


13.7 


13-6 


13-5 


13-3 


10 


15.2 


1 5. 1 


15.0 


14-8 


20 


30.5 


30.3 


30.0 


29.6 


30 


45-7 


45-5 


45.0 


44-5 


40 


61.0 


60.6 


60.0 


^9'?> 


50 


76.2 


75-8 


75.0 


74-1 





88 


88 


87 


6 


8.8 


8.8 


8.7 


7 


10. s 


10.2 


10. 1 


8 


11.8 


II.? 


II. 6 


9 


13-3 


1.3-2 


13.0 


10 


14-7 


14-6 


14.5 


20 


29.5 


29-3 


29.0 


30 


44-2 


44-0 


43-5 


40 


59.0 


58-6 


58.0 


50 


73-1 


73-3 


72.5 





85 


85 


84 


6 


8.5 


8.5 


8.4 


7 


lO.O 


9-9 


9.8 


8 


11.4 


1 1-3 


II. 2 


9 


12.8 


12.? 


12.6 


10 


14.2 


14. 1 


14.0 


20 


28.5 


28.3 


28.0 


30 


42.? 


42.5 


42.0 


40 


57.0 


56-6 


56.0 


50 


71.2 


70.8 


70.0 





82 


82 


81 


6 


8.2 


8.2 


8.1 


7 


9.6 


9.5 


9.4 


8 


II. 


10.9 


10.8 


9 


12.4 


12.3 


I2„I 


10 


13-7 


13-6 


13-5 


20 


27.5 


27-3 


27.0 


30 


41.2 


41.0 


40.5 


40 


55.0 


54 6 


54.0 


50 


68.^ 


68.3 


67.5 



86 

8.6 
10.6 
II. 4 
12.9 

14-3 
28.6 
43-0 
57-3 
71.6 



83 

8.3 

9-7 

II. o 

12.4 
13-8 
27-6 
41.5 
55-3 
69.1 



80 

8.0 

9-3 
10.6 
12.0 

13-3 
26.6 
40.0 

53-3 
66.6 



10 
20 

30 
40 

50 



79 

7-9 
9-3 
10.6 
II. 9 
13.2 
26.5 

39-? 
53.0 
66.2 



2 

0.2 
0.2 
0.2 
0.3 
0-3 
0.6 
i.o 

1-3 
1-6 



I 

o. I 
0.2 
0.2 
0.2 
0.2 
0.5 

o.? 
1.0 

1.2 



P.P. 



i 



356 



TAP>LE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANCiENTS. 

9° 



15 

16 

17 
18 

19 



20 

21 
22 

23 

24 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 

54 




L<nf. Sin. 



d. 



^9 433 
19513 
19592 
19672 
19751 



19 830 
19909 
19988 
20065 

20 145 



20 22 "? 
20 301 
20379 
20457 
20 533 



20 613 
20690 
20 768 
20845 
20 922 



20 999 

21 076 
21 152 
21 229 
21 303 



21 382 
21 458 

21 534 
21 6oq 
21 685 



761 

836 
911 

987 
062 



136 
211 
286 
360 

435 



509 

583 
657 

731 
805 



878 
952 
025 

098 
17T 



244 

317 
390 
462 

535 



607 
679 

751 
823 

89S 



9.23967 



80 
79 
79 
79 
79 
79 
79 
78 
78 
78 
78 
78 
7S 
78 

77 
77 
71 
77 
77 
77 
77 
76 
76 
76 

76 
76 
^6 

71 
76 

75 
75 
75 
7l 
75 
74 
75 
74 
7-+ 
74 
74 
74 
74 
73 
74 
73 
73 
73 
73 
73 

73 
72 

73 
72 
72 
72 
72 
72 
72 
72 

71 



Lo?. Tan. ' 0. d. I Loir. ( df. 



Log. Cos. 



d. 



I9971 
20053 
20 134 
20 216 
.20 297 



.20 378 
20459 
20 540 
20 626 
20 701 



20 781 
20862 
20942 
022 
102 



181 
261 

340 
420 

499 



578 
657 

73? 
814 
892 



21 971 

22 049 
22 127 
22 205 

22 283 



22 366 
22 438 
22.515 

22 593 
22 670 



22747 
22 824 
22 900 
22 977 
23054 



23 130 
23 206 

23 282 

23 358 
23434 



23 510 
23586 
23 661 

23737 
23 812 



23 887 

23 962 
24037 

24 112 
24 185 



24 261 

24335 
24409 

24484 

24558 



24632 



81 
81 
81 
81 
81 
81 
81 
86 
81 
80 
86 
80 
80 
80 

79 
79 
79 
79 
79 
79 
79 
78 
78 
7^ 
78 
7^ 
78 
78 
7^ 
71 
71 
71 
71 
77 

77 
77 
76 
77 
76 
76 

76 
76 

76 

76 

7^ 
7% 
71 
7l 
75 
75 
75 
75 
75 
74 

74 
74 
74 
74 
74 
74 



0.80 02J^ 
0.79947 
0.79865 
0.79784 
0.79703 



0.79 622 
0.79541 
0.79 460 
0.79379 
0.79298 



0.79213 
0.79 138 
0.79058 
0.78978 
0.78898 



0.78 ^l^ 
0.78739 
0.78 659 
0.78 580 
0.78 501 



0.78 422 

0.78 343 
0.78 264 
0.78 186 
0.78 107 



0.78 029 
0.77 951 
0.77 873 
0.77 795 
0.77 717 



0.77 639 
0.77 562 
0.77 484 
0.77 407 
0-77 330 
0.77 253 
0.77 176 
0.77099 
0.77 022 
o. 76 946 



0.76 870 
0.76 793 
0.76 71^ 
0.76641 
0.76 565 



0.76 489 
0.76414 

0.76338 
0.76 263 

0.76 188 



0.76 113 
0.76 038 
0.75963 
0.75888 

0.75813 



0-75 739 
0.75 664 
0.75 596 
0.75 516 
0.75442 
0.75 368 



oer. Cot. : c. d. Lojf. Tiin. 



idtf. Cos. 



99 462 
99 460 
99458 
99456 
99 454 



99452 
99450 
99448 
99 446 
99 444 

99442 
99440 

99:437 
99 435 
99 433 



99431 
99429 
99427 

99425 
99423 



99421 
99419 
99417 
99415 
9 9413 
99 41 1 
99408 
99406 
99404 
99 402 



99 400 
99398 
99 396 
99 394 
99392 



99389 
99387 
99385 
99 383 
99381 



99 379 
99 377 
99 374 
99372 
99370 
99 368 
99 366 

99364 
99361 

99 359 



9 
9 
9 
9 
9 

9 
9 
9 
9 

9 

9-99 335 



99 357 
99 355 
99 353 
99350 
99 348 
99346 
99 344 
99 342 
99 339 
99 33? 



L(»>r. Sin. 



(>0 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
J6 

45 

44 

43 
42 

41 



25 

24 

23 
22 
21 



20 

19 
18 

17 
16 



15 
14 

13 
12 

II 

9 
8 

7 
6 



I', r 





81 


81 


80 


6 


8.1 


8.1 


8.0 


7 
8 


9-5 
10.8 


9-4 
10.8 


9 3 
10.6 


9 


12.2 


12.1 


12. ol 


10 
20 


13.6 

27.1 


135 
27.0 


13.3I 
26.6 i 


30 


40.7 


40.5 


40.0 


40 
50 


54-3 
67.9 


54.0 
67.5 


53-3 
66.6 



79 

7-9 
9.2 

10. § 

II. 8 

I3-J 
26.3 

39-5 
52.6 
65-8 



6 

7 
8 

9 
10 

20 

30 
40 

50 



78 

7-8 

9- 1 
10.4 
II. 8 

131 
26. T 

39-2 

52.3 
65.4 



78 

7-8 
9.1 
10.4 
II. 7 
13.0 
26.0 
390 
52.0 
65.0 



77 

7-7 

9.0 

10.2 

II. 5 
12.8 
25-6 
38.5 
51.3 

64. T 



76 



/ 
8 

9 
10 
20 
30 
40 



7 
8 


6 
9 


10 


2 ! 


1 1 


5 


12 


7i 


25 
38 


5! 
2 


51 
63 




1. 



76 

7.6 

8.8 
10. T 

II. 4 

12.6 

25-3 
38.0 

50.6 
63- 3 



75 

7-5; 

8.^1 
10. o! 
1 1.2 
12.5 
25.0 

37-5 



74 

7-4 

8.6 

98 

1 1. 1 

12.3 

24-6 
37-0 





73 


73 


6 


7-3 


7-3, 


7 


8.6 


8.5! 


8 


9.8 


9-7 


9 


II. 


10.9 


10 


12.2 


12. T 


20 


24.5 


24-3 


30 


36.7 


36.5 


40 


49.0 


48.6 


50 


61.2 


60.8 



50.0 49.3 
62.5 61.6 



72 

7.2 

8.4 
9.6 

10.8 

12.0 
24.0 
36.0 

48.0 

60.0 





n 


71 


5 


2 


6 


7-1 


71 


0.2 


0.2 


7 


8. .3 


8.3 


03 


0.2 


8 


9.5 


9.4 


0.3 


0.2 


9 


10.7 


106 


0.4 


0.3 


10 


II. 9 


II. 8 


0.4 


03 


20 


23-8 


23-6 


0.8 


0.6 


30 


35.7 


35-5 


1.2 


I.O 


40 


47-6 


47.3 


'•6 


1-3 


50 


59.6 


59.1 


2.1 


1-6 



r. i*. 



80' 



357 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

10° 



Log. Sin. d. 



9.23967 
9.24 038 
9.24 1 10 
9.24 181 
9.24 252 



9-24323 
9.24394 
9.24465 

9.24536 
9.24 607 



10 

II 

12 

13 
14 



15 
16 

17 
18 

19 



9.24677 
9.24748 
9.24818 
9.24888 

9-24 958 



9.25 028 
9.25 098 
9.25 167 

9.25237 
9 25 306 



20 

21 
22 
23 
24 



26 

27 
28 

2Q 



9.25 376 

9.25443 
9.25514 

9-25 583 

9 25652 



30 

31 
32 
33 

34 



35 

36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 
54 



55 

56 
57 
58 
59 



60 



9.25 721 
9.25790 

925858 
9.25927 

9-25 995 

9.26 063 
9 26 13T 
9 26 199 
9.26 267 
9-26335 

9. 26 402 
9.26470 
9.26 537 
9. 26 605 

9.26 672 



9.26739 

9.26 805 
9-26873 
9 26 940 

9.27 007 



9.27073 
9.27 140 

9-27205 
9.27 272 

9-27 339 



9.27405 
9.27471 

9-27 536 
9.27 602 

9.27668 



9-27 733 
9-27 799 
9.27 S64 

9-27 929 
9.27995 



9.28 060 
Log. Cos. ' 



71 
7t 
71 
71 
71 
71 
71 
71 
70 

70 

70 

70 

76 

70 

69 

70 

69 

70 

69 

69 

69 

69 

69 

69 

^^^ 
6q 

6§ 

68 
68 

^^ 
68 
68 
68 
67 

^1 
68 

67 
67 
67 
(^1 
67 
^1 
66 
67 

66 
66 
66 
66 

66 
66 
66 

65 
66 

65 
65 
65 
65 
65 
65 
65 



Log. Tan. 



9.24632 
9.24705 
9.24779 
9.24853 
9.24925 



9.25 000 
9.25073 
9.25 146 
9.25 219 
9.25 292 



9.25 365 

9-25437 
9.25 510 
9.25 582 
9.25654 



d. 



9-25 727 

9-25799 
9.25871 

9.25 943 

9.26 014 



9. 26 085 
9.26 158 
9.26 229 
9. 26 300 
9 26 371 



9-26443 
9.26514 
9.26 584 
9.26 655 

9.26 726 



9.26 795 
9.26 867 

9.26 93f 

9.27 007 
9.27 078 



9.27 148 
9.27 218 
9.27 287 

9-27 357 

9.27427 



9.27495 
9.27 566 
9-27635 
9.27704 
9-27 773 



9.27 842 
9.27 91T 

9.27 9S0 

9. 28 049 
9.28 117 



9.28 186 
9.28254 
9.28 322 
9.28 390 
9 28459 



9.28 527 
9.28 594 
9.28 662 
9.28 730 
9.28 79f 



9.28865 



Log. Cot. 



jc^d^ 

73 
74 
73 
73 
73 
11> 
73 
7j 
7Z 

73 

72 

72 

72 

72 

72 

72 

72 

72 

71 

72 

71 

71 

7? 

n 
71 

71 
70 

71 

70 

70 
70 
70 
70 
70 
70 
70 

69 
70 

69 
69 

69 
69 

69 
69 

69 
69 

68 
69 
68 

68 
6Z 

68 
68 

68 
68 

67 
6Z 

^7 



Log. Cot. 



0.75 368 
0.75294 
0.75 220 
0.75 147 
0.75073 



0.75 000 
0.74927 
0.74854 
0.74781 
0.74708 



0.74635 
0.74 562 
0.74490 
0.74417 
0.7434! 



0.74273 
0.74 201 
0.74 129 
0.74057 
0.73985 



0-73913 
0.73 842 

0.73 771 

0.73 699 

0.73628 



0.73 557 
0.73486 

0.73415 
0.73344 
0.73274 



0.73 203 

0.73 J33 
0.73 062 

0.72 992 
0.72 922 



0.72 852 
0.72 782 
0.72 712 
0.72 642 
0.72 573 



0.72 503 
0.72434 
0.72 365 
0.72 295 
0.72 225 



0.72 157 
0.72 088 
0.72 020 
0.71 951 
0.71 882 



0.71 814 
0.71 746 
0.71 677 
0.71 609 
0.71 541 



0.71 473 
0.71 405 

0.71 337 
0.71 270 
0.71 202 



0.71 135 



c. d. I Log. Tan. 



Log. Cos. 



9-99 335 
9-99 333 
9-99330 

9-99 328 
9.99326 



9.99324 
9.99321 

9-99319 
9-99317 
9-99315 



9.99312 
9.99316 
9.99308 
9.99 306 
9-99303 



9.99301 
9.99299 
9.99295 
9.99294 
9.99292 



9.99290 
9.99287 
9.99285 
9.99283 
9.99 280 



9.99278 
9-99276 
9.99273 
9.99271 
9.99269 



9.99265 
9.99264 
9.99 262 
9.99259 
9.99257 



9.99255 
9.99252 
9.99250 
9.99248 
9.99245 



9.99243 
9.99 246 
9.99238 
9.99236 

9-99233 



9.99231 
9.99228 
9.99225 
9.99224 
9.99 221 



9-99219 
9-99 216 
9-99214 
9.99 212 
9.99209 



9.99207 
9.99204 
9 99 202 

9-99 199 
9-99 197 



9 99 194 



00 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 

43 

42 

41 



40 

39 
38 
37 
36 



35 
•34 
33 
32 
31 



30 

29 
28 
27 
26 



25 

24 

23 
22 
21 



p. p. 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 



10 

9 
8 

7 
6 



Log. Sin. 



74 



6 


7.4 


7 


8.6 


8 


98 


9 


II. I 


10 


12.3 


20 


24 6 


30 


37.0 


40 


49-3 


50 


61.6 



73 

7-1 
8.6 
9.8 

II. o 

12.2 

24.5 

36.7 
49.0 
61.2 



73 

7.3 
" 5 
7 

9 
I 

3 
5 
6 



9 
10 

12 

24 
36 
48 
60 





72 


72 


7 


I 


7 


6 


7.2 


7.2 


7-1 


7. 


7 


8.4 


8.4 


8 


3 


8. 


8 


9-6 


9.6 


9 


I 


9- 


9 


10.9 


10.8 


10 


7 


10. 


10 


12. 1 


12.0 


II 


9 


II. 


20 


24.1 


24.0 


23 


8 


23- 


30 


36.2 


36.0 


35 


7 


35- 


40 


48.3 


48.0 


47 


6 


47. 


50 


60.4 


60.0 


59 


6 


59- 



6 


7.6 


7-0 


6.9 j 


7 


8.2 


8.1 


8.1 


8 


9-4 


9-3 


9.2 


9 


10.6 


10.5 


10.4 


10 


11.^ 


11.6 


11.6 


20 


23.5 


23.3 


23.T 


30 


35-2 


35-0 


34-7 


40 


47.0 


46.5 


46.3; 


50 


58.^ 


58.3 


57-91 



6 

7 
8 

9 
10 
20 
30 
40 
50 



70 

7-' 
8. 
9., 
0.1 
I. 

^ 

,5.: 

7-< 
8.' 

68 

6.8 
8.0 
9.1 
10.3 
II. 4 
22.8 
34.2 
45-6 
57.1 

68 



70 

7-' 
8. 

9- 
o. 

i.( 

3- 

>5-' 
.6., 

8., 

68 

6.S 

7-9 
9.6 

10.2 
11-3 

22-6 

34-0 

45-3 

56.6 

66 

e 

7 

c 
c 
c 
,c 
c 
c 

2 



69 

6.C 
8.] 
9.: 
o.z 

I.( 
53.1 

14-^ 
.6.1 

\7-l 
6^ 

6.^ 

7-9 
9.0 

10. 1 

11. 2 

22.5 

33-^ 
45.0 

c6.2 
'6S 



6 

7 
8 


6.6 
7.1 
8.8 


6.6 

7-7 
8.8 


6.5 
7-6 

8.^ 


6. 

7- 
8. 


9 
10 


lO.O 

11. 1 


9.9 
II. 


9.8 
10.9 


9. 
10. 


20 


22.1 


22.0 


21.8 


21. 


30 
40 
50 


33-2 

44- S 
55-4 


33-0 
44-0 

55.0 


32-^ 
43-6 
54.6 


32. 
43. 
54- 



6 

7 
8 

9 
10 
20 

30 

40 

50 





^ 





2 





3 









3 





4 





4 





8 


I 


2 




^ 


1 


6 


2. 


I 



2 

0.2 
0,2 
0.2 

0.3 
0.3 
0.6 

I.O 

1-3 
1-6 



6 
5 
3 
I 

69 

6.9 

8.6 

9.2 

10.3 

II. 5 

23.0 

34-5 
46.0 

57-5 

67 

6.7 
7.Z 

8.9 
10.6 

II. I 

22.3 

33-5 
44-6 
55-8 
65 

5 
6 

6 

7 
8 
6 
5 
3 
I 



P. P. 



79 



358 



T\HLE VII. — LOGARITHMIC SINES, COSINES. TANGENTS. AND COTANGENTS. 

11° 



10 

1 1 

12 

14 



15 
i6 

17 
19 



20 

21 
-J 

^4 



26 

-7 

28 

-9 

ao 



34 



J3 
36 
37 
38 
39 



10 

41 
42 
43 
44 



-15 
46 

47 
4S 
49 

.'>0 

51 
52 

34 

55 
56 

S7 
58 

(>0 



Lotf. sill. 



•I. 



28 060 
28 125 
28 189 
28254 
28 319 



28383 

28448 
28 512 

28576 
28641 



703 

769 

832 

896 
960 



29 
29 
29 
29 
29 



087 
156 
213 

277 



29340 

29403 
29 466 

29528 
29591 



29654 

29716 
29779 
29841 
29903 



29963 
30027 
33089 
30 151 
30213 



30275 
30336 
30398 
30459 



o ;2o 



30582 
30643 
30704 

30765 
30 826 



30886 

30947 
31 008 

068 
129 



189 
249 
309 
370 

429 



489 

549 
609 

669 

728 

Lost. Cos. 



65 
64 
65 
64 
64 
64 
64 
64 
64 
64 
64 
63 
64 
63 
63 
63 
63 
63 
63 
63 
63 

63 
62 

63 
62 
62 
62 
62 
62 
62 
62 
62 
62 
61 
62 
61 
61 
61 
61 
61 
61 
61 
61 
61 
60 

6r 
60 
65 
60 
60 
60 
60 
60 

59 
60 
60 

59 
60 

59 

59 

"d7" 



Loff. Tiiii. c. (1. 1 Loir. int. Lotr. Cos 



28 865 
28932 

29 000 
29 067 
29 134 



29 201 
29 268 

29 jj:) 
29401 

29468 



29535 
29 601 
29 667 

29734 

29 800 

29866" 
29932 
29998 
30064 

30 129 



30195 
30 260 

30326 

30391 
30456 



30 522 

30587 
30652 
30717 
30781 



30846 
30 91 1 

30975 
040 

104 



168 

232 

297 
361 

424 



488 
552 
616 
679 

743 



806 
869 

933 
996 

32059 



-32 I 22 
32185 
32 248 
32310 

32 373 



32436 
32498 
32 566 
32 623 
32 68g 

9-3274^ 

Loir. Cot. 



67 
67 
67 
67 
67 

66 
67 
66 
67 

66 

66 
66 

66 
66 

66 
66 
66 
66 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
64 

65 
64 
64 
64 
64 

64 
64 
64 
64 
63 
64 
64 
63 
63 
63 
63 
63 
63 
63 
63 
63 
63 
63 
62 

63 
62 
62 
62 
62 
62 
62 

77T 



71 135 
71 067 
71 000 

70933 
70 866 



70798 
70732 
70 665 
70 598 
70531 



70465 

70398 
70332 
70 266 
70 200 



70134 
70068 
70 002 

69936 
69 870 



69 805 
69739 
69674 
69 608 
69 543 



69478 

69413 
69348 
69 283 

69 218 



69 153 
69 089 

69 024 

68960 

68896 



68831 
68767 
68703 
68639 
68575 



68 511 
68447 
68384 
68 326 
68257 



68 193 
68 136 
,68067 
.68 004 
'•67 941 



.67 878 
67815 

67752 
67689 
67 626 

.67 564 
.67 501 

67 439 
•67 377 
67 314 
67 252 
otr. Tan. 



9-99 
9.99 

9-99 
9.99 
9.99 



9.99098 
9 99 096 



9.99093 
9.99091 
9.99088 
9.99085 
9-99083 
9.99086 
9.99077 
9.99075 
9.99072 
9-99069 
9.99067 
9.99064 
9.99062 
9.99059 
9.99056 



9.99054 
9.99051 

9-99 04 8 
9.99046 

999043 
9.99046 

I,ou'. sin. 



1'. 1' 



21 



20 

19 
18 

17 
16 





/ 
8 




9 




10 




20 




30 




40 




50 




6g 


6 


6.6 


7 
8 


7-f 
8.8 


9 


lO.O 


10 


II. I 


20 


22.1 


30 


33-2 


40 


44-3 


50 


55-4 




64 


6 


6.4 


7 
8 


7-5 
8.6 


9 


9-7 


10 


10.7 


20 


21.5 


30 


32.2 


40 


43-0 


30 


53-7 




62 


6 


6.2 


7 
8 


7-3 
8.3 


9 


94 


10 


10.4 


20 


20.8 


30 


31.2 


40 


41.6 


50 


52.1 



67 

7-9 
9.0 

10.1 i 

11. 2 ! 
22. ^ 

337 
45-0 
56.2 

66 

6.6 

7-7 
8.8 



67 

6.7 

7.8 

8.9 

lo.o 

II. T 

22.3 

33-5 
44.6 
55-8 

65 

6 



9.9 

II. o 

22.0 
330 
44.0 
55.0 
64 



6 


4 


6.3 


7 


4 


7-4 


8 


5 


8.4 


9 


6 


9-5 


10 


6 


10.6 


21 


3 


21. 1 


32 





31-7 


42 


6 


42.3 


53 




J 


52.9 



62 

6.2 

7.2 

8.2 

9.3 

10.3 

20.6 
31.0 



6 

7 
8 

9 
10 
20 

30 
40 

50 



41 

51 

66 

6.6 

7.6 

8.6 

9.1 

10. 1 

20. T 

30.2 

40.3 
50.4 

3 



9 
10 

21 
32 
43 
54 

63 

6.; 
7-' 
8.. 

9-: 

o.( 
I.' 

I.! 
2. ■ 

2.( 
61 

6.1 

7.2 
8.2 

9-2 
10.2 
20.5 

30- 7 

41.0 

51.2 
60 
6.0 
7.0 
8.0 
9.0 

1 0.0 
20.0 

30.0 

40.0 
50.0 

2 



65 

6.5 

7 ' 
8 



9 
10 

21 

32 
43 
54. 

63 

6.3 
7.3 
8.4 
9-4 
10.5 
21.0 

31-5 

42.0 

52.5 

61 

6.1 

7-1 

8.T 

9.1 

10. 1 

20.3 

30.5 
40.6 
50-8 
59 
5-9 
9 
9 
9 
9 
8 
7 
6 
6 



6 

7 
8 

9 
10 
20 
30 
40 
50 



0-3 


0.2 


0.3 


0.3 


0.4 


0-3 


0.4 


0.4 


0.5 


0.4 


I.O 


0.8 


1-5 


1.2 


2.0 


1-6 


2.5 


2.1 



0.2 
0.2 
0.2 

03 
0.3 

0-6 
1.0 

1-3 
1-6 



v. I'. 



78' 



359 



TABLE VII.— 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 

12° 



10 

II 

12 

13 
14 



liOic. Sin. 



9.31 788 

9-31 847 

9' 3 1 906 
9.31 966 
9.32025 



9.32084 

9-32 143 
9.32 202 
9.32 260 
9-32 319 



15 
16 

17 
18 

19 

20 

21 

22 

23 

24 



9-32378 
9-32 436 
9-32495 
9-32553 
9.32611 



25 
26 

27 
28 

29 

30 

31 
32 
33 
34 



9.32670 
9.32 728 
9.32 7^6 
9.32844 
9.32 902 



9.32 960 

9-33017 
9-3307? 
9-33 133 
9-33 190 



9-33248 
9-33 305 
9-33 362 
9-33419 
9-33 476 



35 

36 
37 
3S 
39 



40 

41 

42 

43 
44 

45 
46 

47 
48 

49 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 
GO 



9-33 533 
9-33590 
9-3364? 
9-33704 
9-33761 



9-3381? 
9-33874 
933930 
9-33 9S7 
9- 34 043 



9.34099 
9.34156 
9.34212 
9.34268 
9-34324 

9-34 379 
9-34 435 
9-34 491 
9-34 547 
9. 34 602 



9.34658 

9-34713 
9-34768 
9.34824 

9-34879 



9-34 934 
9-34989 
9-35044 
9-35099 

9-35 154 



9.35 209 



Log.jCos, 



d. 



59 

59 

59 

59 

59 

59 

59 

58 

59 

58 

58 

58 

58 

58 

58 

58 

58 

58 

58 

58 

57 

58 

57 

57 

5? 

57 

57 

57 

57 

57 

57 

57 

57 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

Si 
56 
5? 
56 
Si 
5? 
5? 
55 
55 
55 
55 
55 
55 
54 
55 
55 



Log. Tail. 



9-32747 
9,32 809 

9.32871 

9-32933 

9-32995 



9-33057 

9-33 118 
9.33 186 

9-33242 

9-33303 



c. d. 



9-33364 
9-33426 

9-33487 
9-33548 
9.33609 



9.33670 

9-33731 
9-33792 
9-33852 
9-33913 



9-33 974 
9-34034 
9-34095 
9-34155 
9-34215 



9-34275 
9-34336 
9-34396 
9-34456 
9-34515 



9-34 575 
9-34635 
9-34695 

9-34 754 
9.34814 



9-34873 

9-34 933 
9. 34 992 

9-35051 

9-35 I TO 



9-35 169 
9-35 228 
9.35287 

9-35 346 
9-35405 



9-35464 
9-35 522 
9-35 581 
9.35640 
9-35698 



9-35 756 
9-35815 
9.35873 
9.35931 
9-35989 



9.3604? 
9.36 105 
9.36163 
9.36 221 
9-36278 



d. 



9-36336 



62 

62 

62 

62 

61 

61 

62 

61 

61 

61 

6i 

61 

61 

6i 

60 

61 

61 

66 

61 

60 

60 

66 

60 

66 

60 

60 

60 

60 

59 
60 
60 

59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
58 
58 
59 
58 
58 

58 
58 
58 
58 
58 
58 
58 
5f 
58 
5? 
58 



Log. Cot. 



0.67 252 
0.67 196 
0.67 128 

o. 67 065 
o. 67 004 



0.66 943 
0.66 881 
0.66 819 
0.66758 
0.66 695 



0.66 635 
0.66 574 
0.66 513 
0.66 452 
0.66 396 



0.66 330 
0.66 269 
0.66 208 
0.66 147 
0.66085 



0.66 026 
0.65 965 
0.65 905 
0.65 845 
0.65 784 



0.65 724 
0.65 664 
0.65 604 
0.65 544 
0.65 484 



0,65 424 
0.65364 
0.65 305 
0.65 245 
0.65 186 



0.65 125 
0.65 067 
0.65 008 
0.64948 
0.64889 



0.64 836 
0.64 771 
o. 64 7 1 2 
0.64 653 
0.64 594 



0.64536 
0.6447? 
0.64418 
o. 64 360 
0.64 302 



0.64243 

0.64 185 
0.64 127 
o. 64 068 
0.64016 



0.63952 
0.63894 

0.63 837 
0.63 779 
0.63 721 



0.63.663 



iO g. Cot, le d. I Log. Tan. 



Log. Cos. 



9.99040 

9-99038 

9-99035 
9.99032 

9.99029 



9-99027 
9.99024 
9.99021 
9.99019 
9.99016 



999013 
9.99 016 
9. 99 008 
9.99005 
9.99 002 



9.98999 
9.98997 
9.98994 
9.98991 
9.98988 



9.98986 

9-98983 
9.98 986 

9-9897? 
9.98975 



9.98972 

9.98969 

9.98965 

9-98963 
9.98 961 



9.98958 
9.98955 
9.98952 

9-98949 
9-98947 



9.98944 
9-98 941 
998938 
9-98935 
9-98933 



9.98930 
9.98 927 

9-98924 
9.98 921 

9.98 9I8 



9-98915 
9.98913 

9.98 910 

9.98907 

9.98904 



9.98 901 
9.98898 

9-98895 
9.98 892 
9.98 890 



9.98887 
9.98884 
9.98881 
9.98878 
9-98875 



9.98 872 



Log. Sin. 



00 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 

23 

22 

21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 



10 

9 
8 

7 
6 



P. P. 





62 


61 


61 


6 


6.2 


6.T 


6.1 


I 


7.2 
8.2 


7.2 
8.2 


7-1 
8.1 


9 
10 


9.3 
10.3 


9.2 
10.2 


9-1 
10. 1 


20 
30 
40 
50 


20.6 
31.0 

41-3 
51-0 


20.5 

30.7 
41.0 
51.2 


20.3 

30.5 
40.6 
50.8 





66 


60 


59 


59 


6 


6.6 


6.0 


5 9 


5-9 


7 


7.6 


7.0 


6.9 


6.9; 


8 


8.6 


8.0 


7.9 


7-8 


9 


9.1 


9.0 


8.9 


8-8 


10 


10. 1 


10.0 


9.9 


9-8 


20 


20.1 


20.0 


19-8 


19-6 


30 


30.2 


30.0 


29.7 


29.5 


40 


40.3 


40.0 


39-6 


39.3 


50 


50.4 1 


50.0 


49-6 


49.1 



6 

7 
8 

9 
10 

20 
30 
40 
50 



58 

5-8 
6.8 

9-? 

19-5 
29.2 

39-0 
48.? 



58 

5-8 
6.? 

7-? 

^.7 

9-6 

19-3 

29.0 

38.6 
48.3 



Si 

5-? 
6.7 

7-6 
8.6 
9.6 

19.1 
28.? 

38.3 
47-9 



57 

5-7 
6.6 
7.6 
8.5 

9-5 
19.0 
28.5 
38.0 

47.5 





5S 


56 


5S 


55 


6 
7 


5-6 
6.6 


5-6 
6.5 


5-5 

6.5 


5-5 

6.4 


8 
9 


7.5 
8-5 


7.4 
8.4 


7-4 
8.3 


7-3 
8.2 


10 

20 


9-4 
18.8 


9-3 
18.6 


9-2 
18.5 


9-1 

18.3 


30 


28.2 


28.0 


27.? 


27.5 


40 
50 


37.6 
47.1 


37.3 
46.6 


37-0 
46.2 


36.6 
45-8 





54 


3 


6 


S-4 


0.3 


7 


6. .3 


0.3 


8 


7.2 


0.4 


9 


8.2 


0.4 


10 


9-1 


0-5 


20 


18. 1 


I.O 


30 


27.2 


1-5 


40 


36.3 


2.0 


50 


45-4 


2.5 



2 

0.2 

0.3 
0.3 
0.4 
0.4 
0.8 

1.2 

1-6 
2.1 



p. p 



77' 



360 



TABLE VII. — LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS, 

13" 



26 
27 
28 
29 



30 

31 
32 
33 

34 



35 
36 
37 
38 
39 



40 

41 

42 
43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 

54 



55 
56 
57 
58 
59 



60 



iO:;. Sill. 



(I. 



9.35 209 

9-35 263 
9-35 318 
9-35 372 
9-35427 



9-35481 
9-35 536 
9-35 590 
9-35 644 
9-35698 



9-35 752 
9-35805 
9.35 865 

9-35914 
9.35968 



9.36 021 
9.36075 
9.36 123 
9.36 182 
9-36235 



20 


9.36 289 


21 


9-36342 


22 


9-36393 


23 


9-36448 


24 


9.3650? 



9- 36 554 
9. 36 607 
9. 36 660 

9 36713 
9-36766 

9-36818 
9.36 871 

9-36923 
9-36976 
9-37 028 



9.37081 
9-37 133 
9-37 185 
9-3723^ 
9.37 289 



9-37341 
9 37 393 
9-37 445 
9-37 497 
9-37 548 



9. 37 600 
9-37652 
9-37703 
9.37755 
9-37 806 



9-37857 
9.37909 
9.37960 
9.3801T 
9.38 062 



9.38 113 
9.38 164 
9.38215 
9.38 266 
9-38317 



9-38367 



54 

5-4 

5-4 

54 

54 

54 

54 

54 

54 

54 

54 

54 

53 

54 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

52 

53 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 
52 
51 

52 
51 
52 
51 
51 
51 
51 
51 
51 
51 
51 
51 

51 
51 
50 
51 
51 

50 



Ian. 



<-. (1. 



( (»t. 



9-36336 
936394 

9.36451 
9.36509 

9-36566 



9.36623 

9. 36 68 1 

9-36738 

9-36795 
9.36852 



9.36909 
9. 36 965 
9.37023 
9.37080 

9-37 136 



9-37 193 
9.37 250 

9-37 306 
9- 37 363 
9.37419 



Log. Cos. I (1. 



9-37 475 
9-37 532 
9.37 588 

9-37644 
9-37 700 



9-37 756 
9.37 812 

9-37868 

9-37924 

9-37 979 



9-38035 
9.38091 

9-38 146 
9.38 202 
9-38257 

9-38313 
9-38368 

9-38423 
9-38478 
9-38 533 



9-38 589 
9- 38 644 
9-38698 

9-38753 
9-38 808 



9.38863 
9.38 918 
9-38972 
9.39027 
9.39081 



9-39 136 
9.39 190 

9- 39 244 
9-39299 
9-39 353 



9-39407 
9-39461 
9-39 5i§ 
9-39569 
9-39623 



9-39677 

Log. Cot. 



57 
57 
57 
57 
57 
57 
57 
57 
57 
57 
57 
56 
57 
56 
57 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 

Si 
56 
56 
5^^ 
56 
5S 
Si 
55 
Si 
Si 
55 
55 
55 

55 
-? 
:>D 

55 
54 
55 
55 
54 
55 
54 
5^ 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 



0.63 663 
0.63 606 
0.63 548 
0.63491 
063433 



0-63 376 
0.63319 
0.63 262 
0.63 204 
0.63 147 



0.63 096 
0.63033 
0.62 977 
0.62 920 
0.62 863 



0.62 806 
0.62 730 
0.62 693 
0.62 637 
0.62 580 



l.dU'. Cos. 



0.62 524 
0.62 468 
0.62 412 
0.62 356 
0.62 299 



0.62 243 
0.62 188 
0.62 132 
0.62 076 
0.62 020 

06 



0.6 
0.6 
0.6 
0.6 

0.6 
0.6 
0.6 
0.6 
0.6 



c. d. 



0.6 
0.6 
0.6 
0.6 
0.6 



0.6 
0.6 
0.6 



964 
909 

853 
798 
742 
687 
632 

576 
521 

466 



411 

356 
301 

246 
191 



137 
082 

7 



0.60973 
0.60 913 



0.60 8 64 
0.60 809 
0.60 755 
0.60 701 
0.60 647 



0.60 592 
0-60 538 
0.60 484 
0.60 430 
0-60375 

0.60 323 

Log. Tan. 



9.98 872 
9. 98 869 
9.98865 
9-98863 
9.98 860 



9.98 858 
9.98855 
9.98852 
9.98849 
9.98 840 

9-98843 
9.98 840 
9-98837 
9.98834 
9.98831 



9.98 828 
9.98 825 
9.98822 
9.98 819 
9.98815 



9.98813 
9.98 816 
9.98 S07 
9.98 804 
9.98 80T 

9-98798 
9.98795 

9.98792 

9-98789 
9.98 786 



9-98783 
9.98 780 

9-98777 
9-98774 
9.98771 



9.98768 

9-98765 
9.98 762 
9.98759 
9-98755 



9.98752 
9.98749 
9.98745 

9-98743 
9.98 746 



998737 
9-98734 
9.98731 
9.98728 
9-98725 



9.98 72T 

9-98718 
9.98/ 15 
9.98 712 
9-98 709 



9.98 706 

9-98 703 
9.98 700 
9.98695 
9.98693 



9.98 696 



40 

39 
3^ 
37 



I'. !'. 



-3 
24 

23 
22 

21 



20 

19 
18 

17 
_i6 

15 
14 

13 
12 
1 1 

To 

9 

8 
7 



Loer. Sin. 



6 

7 
8 

9 
loj 

20 i 

30' 
401 

50 I 



57 

S-7 
6.7 

7-6 
8.6 
9.6 
19.T 
28.^ 
38.3 
47-9 



57 

5-7 



6 

7 
8 

9 
19 
28 

38 

47 



56 

5-^ 
6. 

7- 



18 
28 
37 
47 



•0 
.6 


3- 

6. 


•5 

•5 


7- 
8. 


•4 


9- 


•8 


18. 
28. 


•6 
.1 


37- 
46. 





55 


55 


54 


6 


5-5 


5-5 


5--+ 


7 


6.5 


6 


4 


6.3 


8 


7.4 


7 


3 


7.2 


9 


8.3 


8 


2 


8.2 


10 


9.2 


9 


I 


9-1 


20 


18.5 


18 


3 


18.1 


30 


27-7 


27 


5 


oy n 


40 


37.0 


36 


6 


36.3 


50 


46.2 


45 


8 


45-4 



56 

6 



54 

5-4 

6.3 

7-2 

8.1 

9.0 

18.0 

27.0 

36.0 

45.0 





53 


53 


52 


52 1 


6 

7 


5-3 
6.2 


5-3 
6.2 


5-2 
6.1 


5-2 
6.6 


8 


7.1 


7.0 


7.0 


6.9 


9 
10 


8.0 
8.9 


7-9 
8-8 


7-9 
^-7 


7.8 
8.6 


20 
30 


17.8 
26. f 


17-6 
26.5 


17.5 
26.2 


26.0 


40 


35.6 


35-3 


35.0 


34-6 


50 


44.6 


44.1 


43-7 


43.3 



6 

7 
8 

9 
10 

20 
30 
40 
50 



51 

5.1 
6.0 

6-8 

7-7 

8.6 

17.1 

25. f 

34-3 
42.9 



6 

7 
8 

9 
10 

20 

30 
40 

50 



3 

0.3 
0.4 
0.4 
0.5 
0.6 
I.I 
i.f 
2.3 
2.9 



51 

5-1 
5-9 
6.8 

7-6 

8-5 
17.0 

25-5 
34.0 
42.5 



0-3 

0.3 
0.4 
0.4 
0.5 
i.o 

1.5 
2.0 

2.5 



50 

5.0 



5 
6 

7 

8 

16 

25 
33 
42 



2 

0.2 

0.3 
0.3 
0.4 
0.4 

0.8 
1.2 

1-6 
2.1 



1 . 1 



76° 



361 



TABLE Vli.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

14° 







_9_ 
10 
II 

12 

13 

14 



15 
16 

18 
19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 



30 

31 

32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 
43 
44 

45 
46 

47 
48 

49 



50 

51 

52 
53 
Ji 
55 
56 
57 
58 
59 



60 



Log. Sill. d. 



9-3836? 
9.38418 

9-38468 

9.38519 
9.38569 



9.38 620 
9.38670 
9.38 726 
9.38771 
9.38821 



9.38871 

9.38 921 
9.38971 

9.39 021 
9.39071 



9.39 120 
9.39176 
9.39 220 
9.39269 

9-39319 



9-39368 
9.39418 

9-3946? 

9-39515 
9.39566 



9.39615 
9.39664 

9-39713 
9.39762 

9-39 81 1 



9. 39 860 
9.39909 

9-39 95? 
9.40006 
9.40055 



9.40 103 
9.40152 
9. 40 200 
9.40249 
9.40 297 



9-40345 
9-40394 
9.40442 
9.40490 
9.40 538 



9.40 586 
9.40634 
9.40682 
9.40 730 
9.4077? 



9.40 825 
9.40873 
9.40 920 
9.40968 
9.41015 



9.41 063 
9.41 116 
9.41 158 
9.41 205 
9.41 252 



9.41 299 



Log. Cos. d. 



50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 

49 
50 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
48 
48 
49 

48 

48 
48 
48 
48 

48 

48 
48 

48 
48 

48 
48 
48 
48 
47 
48 

4? 
4? 
4? 
4? 
4? 
4? 
4? 
47 
4? 
47 



Log. Tau. 



9-39677 
9-39 731 
9-39784 

9-39838 
9.39892 



9-39 945 

9-39 999 
9.40052 
9.40 106 
9.40159 



9. 40 2 1 2 
9.40 265 

9-40 318 
9.40372 

9-40425 



9.40478 
9.40531 
9.40 583 
9-40636 
9.40 689 



9.40742 
9.40794 
9.40847 
9.40899 
9.40952 



9.4 
9-4 
9-4 
9-4 
9-4 



9-4 
9-4 
9.4 

9-4 
9-4 



9-4 
9.4 

9-4 
9-4 
9.4 



9.4 

9-4 
9.4 
9.4 
9.4 



004 
057 
109 
161 
213 



266 
318 
370 
422 

474 



525 
57? 
629 
681 
732 



784 
836 
887 

938 
990 



9.42041 
9.42 092 
9-42 144 
9-42 195 
9-42 246 



9.42 29? 

9-42 348 
9.42 399 
9.42450 
9.42 501 



9.42 552 
9.42 602 
9.42653 

9-42 704 
9.42 754 



9.42 805 



Loe. Cot. 



c. d. I Log. Cot. 



54 
53 
54 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
52 
53 
52 

53 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 

51 
52 
52 
51 
51 
51 
52 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
50 

51 
50 
51 
53 
50 
50 



c. d. 



0.60 323 
0.60 269 
0.60 21 5 
0.60 1 61 
0.60 108 



0.60054 
0.60001 

o. 59 94? 

0.59894 
0.59 841 



0.5978? 

0.59734 
O.5968T 
0.59 628 
0.59575 



0.59 522 
0.59469 
0.59 4I6 
0.59363 
0-5931 I 



0.59258 
0.59 205 

0.59153 
0.59 100 

o. 59 048 



0.58995 
0.58943 
0.58891 
0.58838 
0.58786 



0.58734 
0.58682 
0.58 630 
9.58578 
0.58 526 



0.58474 
0.58 422 
0.58 370 
0.58319 
0.58 26? 



0.58 216 
0.58 164 
0.58 112 

o. 58 061 

O.58CIO 



0.57 958 
0.5790? 
0.57856 
0.57805 

0.57753 



0.57 702 

0.57651 
0.57 606 
0.57549 
o. 57 499 



0.57448 

0.5739? 
0.57 346 
0.57 296 

o 57245 
o'S7 195 

Log. Tan. 



Log. Cos. 



9.98 696 
9.9868? 
9.98684 
9.98681 
9.98678 



9.98674 
9.98 67T 
9.98668 
9.98 665 
9.98 662 



9-98658 
9.98655 
9.98 652 
9.98649 
9. 98 646 



9.98 642 
9.98639 
9-98636 
9.98633 
9-98630 



9.98626 
9-98623 
9.98 620 
9.98 617 
9.98613 



9.98 616 
9.98 607 
9.98 604 
9.98 606 
9.98 59? 



9.98 594 
9.98 591 
9.98 58? 
9.98 584 
9.98 581 



9.98 578 

9.98 574 
9.98571 
9.98 568 
9.98 564 



9.98 561 
9.98558 
9.98 554 
9.98551 
9.98 548 



9.98 544 
9.98 541 
9.98 538 
9-98 534 
9.98 531 



9.98 528 
9.98 524 
9.98 521 
9.98518 
9.98 514 



9.98511 
9.98 508 
9.98 504 
9.98 501 
9.98498 



9-98494 



Log. Sin. 



d. 



00 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 

41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 

23 

22 

21 



20 

19 
18 

17 
16 



15 

14 

13 
12 

II 



10 

9 

8 

7 
6 



p. P. 



6 

7 
8 

9 
10 

20 
30 
40 
50 



54 


53 


5.4 


5-3 


6.3 


6.2 


7.2 


7.1 


8.1 


8.0 


9.0 


8.9 


18.0 


17.8 


27.0 


26.? 


36.0 


35-6 


45-0 


44.6 



53 

5.3 
6.2 

7.0 
7-9 
8.8 

17-6 
26.5 

35-3 
44.1 





52 


52 


51 


51 


6 


5.2 


5.2 


5.1 


5. 


7 


6.1 


6.6 


6.0 


5- 


8 


7.0 


6.9 


6.8 


6. 


9 


7.9 


7.8 


7-7 


7- 


10 


8.? 


8.6 


8.6 


8. 


20 


17.5 


17.3 


17.! 


17. 


30 


26.2 


26.0 


25-? 


25. 


40 


35.0 


34.6 


34.3 


34. 


50 


43-? 


43.3 


42.9 


42. 





56 


SO 


49 


49 


6 


5.S 


5.0 


4.9 


4- 


7 


5-9 


5-8 


5.8 


5- 


8 


6.7 


6.6 


6.6 


6. 


9 


7.6 


7.5 


7.4 


7- 


10 


8.4 


8.. 3 


8.2 


8. 


20 


16.8 


16.6 


16.5 


16. 


30 


25.2 


25.0 


24.? 


24. 


40 


33.6 


33-3 


33.0:32. 


50 


42.1 


41-6 


41.2 


40. 





48 


48 


Al 


47 


6 


4-8 


4.8 


4-? 


4- 


7 


5-6 


5.6 


5-5 


5- 


8 


6.4 


6.4 


6.3 


6. 


9 


7.3 


7.2 


7.1 


7. 


10 


8.1 


8.0 


7.9 


7. 


20 


16. 1 


16.0 


15.8 


15. 


30 


24.2 


24.0 


23-? 


23. 


40 


32.3 


32.0 


31.6 


31. 


50 


40.4 


40.0 


39.6 


39. 



6 

7 
8 

9 
10 

20 
30 
40 
50 



3 

0.3 
0.4 
0.4 

0.5 
0.6 
I.I 
I.? 
2.3 
2.9 



3 

0.3 

0.3 
0.4 

0.4 

0.5 

i.o 

1.5 
2.0 

2.5 



p. P. 



75* 



362 



TABLE VII. — L()GARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

15° 



10 

II 

12 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



^3 

26 
27 
28 
29 



30 

31 
32 
33 

34 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



Lotf. Sill. 



9-4 
9.4 
9.4 
9.4 
9-4 



9.4 

9-4 
9.4 

9-4 
9.4 



9 4 
9-4 
9.4 
9.4 

9.4 



299 

346 
394 
441 
488 



534 
581 
628 
675 
721 



768 
815 
861 
908 
954 



9.42 000 
9.42 047 
9.42093 
9.42 139 
9.42 185 



9.42 232 
9.42 278 
9.42 324 
9.42 369 
9.42415 



9.42 461 
9.42 507 
9-42 553 
9-42 598 
9.42644 



9.42 690 

9-42735 
9.42781 

9.42825 
9.42 871 



9.42917 
9.42 962 
9.4300^ 
9.43052 
9.43098 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



GO 



9-43 143 
9.43 188 

943233 
9.43278 

9-43322 



9-43 367 
9.43412 

9-43 457 
9-43 501 
9-43 546 



9.43 591 

943635 
9.43680 

9-43724 
9-43 768 



9-43813 
9-43857 
9-43 901 
9-43 945 
9-43989 



9-44034 

Log. Cos. 



47 
47 

47 
47 

46 
47 
47 
46 
46 
47 
46 
46 
46 
46 
46 

46 
46 
46 
46 

46 
46 
46 

45 
46 

46 
46 

45 
45 
46 

45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 

45 
44 
45 
44 
4-+ 

45 
44 
44 
44 
4-1 
44 
44 
4-1 
44 
44 
44 



Lou'. Tnii. c. d. 



9.42 805 
9.42 856 
9.42 906 

9-42956 
9.43007 



9-43057 
9 43 107 
9-43 157 
9.43 208 
9.43258 



9-43308 
943358 
9-43408 

9-43458 
9-43 508 



9-43 557 
9.43607 

9-43657 
9-43 706 
9-43 756 



9-43 806 
943855 
9-43 905 
9-43 954 
9-44003 



9-44053 
9.44 102 

9-44 151 
9.44 200 

9-44249 



9.44299 
9-44 348 
9-44 397 
9.44446 

9-44 494 



9-44 543 
9-44 592 
9.44641 
9.44690 
9-44 738 



944787 

9.44835 
9.44884 

9-44932 
9.44981 



9.45029 
9.4507^ 
9.45 126 
9.45 174 
9.45 222 



9.45 270 
9-45 318 
9-45 367 
9.45415 

9:_45J:^ 
9.45 515 

9-45 558 
9-45 606 
9.45654 
9-45 702 



9-45 749 

Log. Cot. 



51 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 

49 
50 
49 
49 
50 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
48 

49 
49 
48 
49 

48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 

48 
48 
48 
47 
48 
48 

^1 
48 

47 



Loe. Col. 



0.57 195 
0.57 144 
0.57094 
0.57043 
o. 56 993 



o. 56 942 
0.56 892 
o. 56 842 
0.56 792 
o. 56 742 



0.56 692 
o. 56 642 
0.56 592 

0.56 542 

o. 56 492 



0.56442 

o. 56 392 

0.56 343 
0.56 293 
o. 56 243 



0.56 194 
0.56 144 
0.56095 
0.56 04^ 
0.55996 



0.55947 
0.55898 

0.55848 

0.55799 
0.55 750 



0.55701 
0.55652 
0.55 603 

0.55 554 
0.55 505 



0.55 456 
0.5540^ 

0.55359 
0.55310 

0.55 261 



liOir. Cos. 



0.55 213 
0.55 164 
0.55 116 
0.55 067 
0.55019 



0.54970 
0.54 922 

0.54874 
0.54 825 

0.5477? 



0.54729 

o. 5468T 

0.54633 
0.54585 
0.54537 



o. 54 489 
0.54441 

0.54393 
0.54346 

o 54 298 

0.54256 



9-98494 
9.98491 

9-98487 
9.98484 
9. 98 481 

9-9847? 
9.98 474 
9-98470 
9.98467 
9.98464 



9. 98 466 
9-98457 

9-98453 
9.98 450 

9-98446 



9-98443 
9-98439 
9-98436 

9-98433 
9.98 429 



9.98 426 
9.98 422 
9.98419 
9.98415 
9.98 412 



9.98 408 
9.98405 
9.98 401 
9-98 398 
9-98 394 



9.98 391 

9-98 387 
9.98 384 
9.98 386 
9.98 377 

998373 
9-98370 
9.98365 
9-98363 
9-98 359 



9.98356 
9.98352 
9-98348 
9-98345 
9-98 341 



c. (1. I Lotr. Tan. 



9-98 338 
9-98334 

9-98331 
9.98 32^ 
9.98 324 



9.98 320 
9-98 3I6 
9-98313 
9-98309 
9-98 306 



9.98 302 
9.98298 
9-98295 
9.98 29T 
9.98288 

9-98 284 
Loir. Sin. 



40 

39 
38 
37 
36 



35 
34 
33 



r. I' 



30 

29 

28 

27 
26 

25 

24 

23 
22 

21 



20 

19 
18 

17 
16 



15 
14 

13 
12 

1 1 



10 

9 
8 

7 
6 





50 


5< 


D 


6 


5.6 


5.0 


7 


5-9 


5 


8 


8 


6.? 


6 


6 


9 


7.6 


7 


5 


10 


8.4 


8 


3 


20 


16.8 


16 


6 


30 


25.2 


25 





40 


33-6 


33 


3 


50 


42.1 


41 


6 



6 

7 
81 

9! 

10, 



49 

4.9 
5-8 
6.6 

7-4 
8.2 



20 16.5 
30,24.? 
4033-0 
5041.2 



47 

4.? 

5 " 
6 



9 
10 

20 15 

3o|23 

4031 

50:39 



49 

4-9 

5 
6 , 

5 

T 
16.3 

24.5 

32.6 
40 " 



47 

4-7 



48 

4 ' 
5 



48 

4-8 

5-6 

6.4 

7.2 

8.0 

16.0 

24.0 

32.0 

40.0 



7 
7 
15 
?|23 
631 
6|39 



46 

4-6 



'5 

23 

31 

TI38 



46 
4.6 

5-3 
6.T 

6.9 

7-6 

15-3 

2123.0 

030.6 
?!38.3 



6 

7 
8 

9 
10 

20 



45 

4.5 
5-3 
6.6 
6.8 
7.6 

15.1 
30 22.^ 
40' 30. 3 

50:37.9 



10 
20 



45 

4-5 
5-2 
6.0 
6.? 

7.5 
15.0 

22. 5 



44 

4-4 
5 



30.0 29 

37.5I37 



40 
50 



0.4 


0.3 


0.4 


0.4 


0.5 


0.4 


0.6 


o.S 


0.6 


0.6 


1-3 


i.i 


2.0 


I-? 


2-6 


2-3 


3.3 


2.9 



9 

7 
4 

8 ^^ 
2 22 

6*29 
M36 



3 

0.3 
0.3 
0.4 
0.4 
0.5 
i.o 

1-5 
2.0 

2-5 



44 

4-4 
5 ' 
5 
6 

7 



74° 



563 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 

16° 



10 

II 

12 

13 
14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 

24 



25 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 

54 



55 
56 
57 
58 
59 



GO 



hog. Si:i. 



9-44034 
9-44078 
9.44122 
9.44 166 
9.44209 



(1. 



9-44253 
9.44 297 

944341 
9-44384 

9-44 428 



9.44472 

9-44515 
9-44 559 
9.44602 
9.44646 



9.44689 

9-44732 
9.44776 
9.44819 
9.44 862 



9-44 905 
9-44 948 
9.44991 

9-45 034 
9.45077 



9.45 120 
9.45 163 
9.45 206 
9.45 249 
9.45 291 



9-45 334 
9-45 377 
9.45419 
9.45462 
9-45 504 



9-45 547 
9.45 589 
9-45631 
9.45 674 
9.45716 



9-45 758 
9.45 800 

9-45 842 
9.45 885 
9.45927 



9-45 969 
9.46 on 
9.46052 
9.46094 
9-46 136 



9.46 178 
9.46 220 
9.46 261 
9-46 303 
9-46 345 



9.46 385 
9.46 428 
9.46469 
9.46 511 
9.46552 



9-46 593 
Lo?. Cos. 



44 
44 
44 
43 
44 
44 
43 
43 
44 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
42 
43 
43 
42 

42 

43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 

41 
42 
42 

41 
42 
41 
41 
42 

41 
41 
41 
41 
41 
41 
d. 



Lost. Tan. c. d. 



9-45 749 
9-45 797 
9-45 845 
9.45 892 
9.45 940 



9.45 98^ 
9.46035 
9.46082 

9.46 129 
9.46 177 



9.46 224 
9.46 271 

9-46318 
9.46 366 
9.46413 



9. 46 460 
9.46 507 
9.46 554 
9.46 601 
9.46647 



9.46 694 
9.46 741 
9.46788 
9.46834 
9.46881 



9.46 928 
9.46974 

9.47 021 
9.47 067 
9.47 114 



9.47 166 
9 47 207 
9.47253 
9-47 299 
9 47 345 



9-47 392 
9.47 438 
9.47 484 

9-47 530 
9-47 576 



9.47 622 
9-47 668 

947714 
9.47 760 
9.47 806 



9-47851 
9.47 897 

9-47 943 
9.47 989 
9.48034 



9.48 080 
9.48 125 
9.48 171 

9.48 216 

9.48 262 



9-48 307 

948353 
9.48 398 

9-48443 

9-48488 

9-48 534 

hog. Cot. 



48 
47 
47 
4? 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
46 
47 
47 
46 
46 
47 

46 
46 
46 
46 
46 
46 

46 
46 

46 
46 

46 
46 
46 

46 
46 
46 
46 

45 
46 
46 

45 
46 

45 

46 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

— 
c. d. 



Log. Cot. 



54256 
54202 

155 
107 
060 



54 
54 
54 



0.54012 
0.53965 
0.53917 
0.53876 
0.53823 



0.53776 

0.53728 
0.53681 

0.53634 
0.53 587 



0.53 
0.53 
0.53 
0.53 
0.53 



540 

493 
446 
399 
352 



0.53 
0.53 
0.53 
0.53 
0.53 



0.53 
0.53 
0.52 

0.52 
0.52 



305 
258 
212 

165 

iii 
072 

025 
979 
932 

886 



0.52 
0.52 
0.52 
0.52 
0.52 



839 
793 
747 
706 
654 



0.52 
0.52 
0.52 
0.52 
0.52 



608 
562 
516 
469 
423 



0.52 
0.52 
0.52 
0.52 
0.52 



377 
33? 
286 
240 
194 



0.52 
0.52 
0.52 
0.52 
o.5t 
0.51 
0.51 
0.51 
0.51 
0.51 



148 

102 

057 

01 1 

96|_ 

920 

874 
829 

783 
738 



0.51 
0.51 
0.51 
0.51 
0.51 



692 

647 
602 

556 
511 



0.51 466 

hog. Tan. I 



Loar. Cos. 



9.98 284 
9.98 286 
9.98277 
9.98273 
9.98 269 



9.98 266 
9.98 262 
9.98258 
9.98255 
9.98 251 



9.98 247 

9-98 244 
9.98 246 

9.98236 
9-98233 



9.98 229 
9.98 22^ 
9.98 222 

9-98218 
9.98 214 



9.98 211 

9.98 207 

9.98203 

9.98 200 

9.98 

9.98 

9.98 



9.98 
9.98 
9-98 



9.98 
9.98 
9.98 
9.98 
9-98 



9.98 
9.98 
9.98 
9.98 
9.98 



9.98 
9.98 
9.98 
9.98 
9.98 



9.98 

9-98 
9.98 
9.98 
9.98 



96 
92 



85 
81 

77 



73 
70 
66 
62 
58 



55 
51 

47 

43 
40 



36 
32 
28 
24 
21 



17 

13 
09 

05 
02 



9.98 098 
9.98094 
9.98 096 
9.98086 
9.98 082 



9.98079 
9.98075 
9.98 071 
9.98 o6f 
9.98063 



9-98059 

liOir. Sin. 



3 
3 
3 
4 

3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 

3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 

3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
"(iT 



00 

59 
58 
57 
5^ 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



25 
24 

23 
22 
21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 



10 

9 

8 

7 
6 



V. V. 





48 


4^ 


6 


4.8 


A- 7 


7 


5.6 


5.5 


8 


6.4 


6.3 


9 


7.2 


7-1 


10 


8.0 


7-9 


20 


16.0 


15-8 


30 


24.0 


23-7 


40 


32.0 


31-6 


50 


40.0 


39.6 





46 


46 


4S 


6 


4-6 


4.6 


4-5 


7 


5-4 


5-3 


5-3 


8 


6.2 


6.T 


6.6 


9 


7.0 


6.9 


6.8 


10 


7-9 


7-6 


7.6 


20 


15-5 


15-3 


15.1 


30 


23.2 


23.0 


22.7 


40 


31.0 


30.6 


30.3 


50 


38.^ 


38.3 


37.9 





44 


43 


43 


6 


4.4 


4.3 


4. 


7 
8 

9 


5-1 
5-8 
6.6 


5-1 
5-8 
6.5 


5- 
5- 
6. 


10 


7-3 


7.2 


7- 


20 


14-6 


14.5 


14. 


30 


22.0 


21. f 


21. 


40 
50 


29-3 
36.6 


29.0 
36.2 


28. 

35- 



47 

4-7 

5-5 
6.2 

7.6 

7-8 
15-6 
23-5 
31-3 
39-1 

45 

4.5 
5.2 

6.0 

7.5 
15.0 
22.5 
30.0 

37.5 



6 

7 
8 

9 
10 
20 

30 
40 

50 



4 

04 
0.4 
0.5 
0.6 
0.6 

1-3 
2.0 

2-6 
3-3 



P. I' 



3 

0.3 
0.4 
0.4 
0.5 
0.6 
i.i 
i-'7 
2.3 
2.9 





42 


42 


41 


41 


6 


4.2 


4.2 


4.1 


4.1 


7 

8 

9 


4.9 

5-6 
6.4 


4-9 
5.6 

6.3 


4.8 

5-5 
6.2 


4.8 

5-4 
6.T 


10 


7-1 


7.0 


6.9 


6.8 


20 


14. 1 


14.0 


13-8 


13-6 


30 
40 


21.2 
28.3 


21.0 
28.0 


20.^ 
27.6 


20.5 
27-3 


50 


35-4 


35-0 


34-6 


34-1 



364 



TAHLE VII. — LOGARITHMIC SINES, COSINES, TANGENTS, AND CO'lAN(iEXTS. 

17" 



4 

5 
6 

7 
8 

9 
10 

II 

12 

13 
U 

15 
i6 

17 
i8 

19 



20 

•21 
22 

23 

24 
25 

26 

28 



30 

31 
32 
jj 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



55 
56 
57 
58 
59 

(;o 



Lotf. Sill. 



(I. 



9-46 593 
9.46635 
9.46676 
9.46717 
9-46758 



9.46799 
9.46 840 
9.46881 

9.46 922 
9- 46 963 
9.47004 

9.47 04^ 
9.47086 
9.47 127 
9.47 168 



9.47 208 
9.47 249 
9.47 290 
9-47 330 
9 47 371 



9.47 41 1 
9.47452 
9.47492 
9-47 532 
9-47 573 



9.47613 

9-47653 
9.47694 

9-47 734 
9 47 774 



9.47814 
9.47 854 
9.47 894 
9-47 934 
9-47 974 



9.48 014 
9.48054 
9.48093 
948 133 
9- 48 173 



948213 
9.48252 
9.48 292 

948 331 
9.48371 



9.48 410 
9.48450 
9.48 489 
9.48 529 
9.48 568 



9.48 607 
9.48646 
9,48686 
9.48725 
9.48 764 



9.48 803 
9.48 842 
9.48 881 
9.48 920 
9.48959 



9.48998 



Loe. Cos. 



41 

4i 
41 
41 

41 
41 
41 
4f 
•41 
41 
41 

40 

41 
40 
40 

41 

40 

40 

40 
46 
40 
40 
40 
40 
40 
46 
40 
43 
40 
4^ 
40 
40 
40 
40 
40 

39 
40 

39 
40 

39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
29 
39 
38 



Loif. Tan. 



9-48 534 
9.48 579 
9.48 624 
9.48 669 
9.48714 



9.48759 
9.48 804 
9.48 849 
9.48 894 
948939 



9.48984 
9.49028 

9-49073 
9.49 118 
9.49 162 



9.49 207 
9-49252 
9.49296 
9-49 341 
949385 



9-49430 
9-49 474 
9-49 518 

9-49 563 
9.49607 



9-49651 
9.49695 
9.49 740 
9.49784 
9.49 828 



9.49872 

9.49 916 
9.49960 

9. 50 004 
9. 50 048 



9. 50 092 
9.50136 
9.50179 
9.50223 
9.50 267 



9.50 31 1 

9-50354 
9.50398 
9.50442 
9.50485 



0. d. 



iOir. Cot. 



9.50529 
9-50572 
9.50616 
9.50659 
9.50702 



d. 



9. 50 746 

9.50789 
9.50832 
9.50876 
9.50919 



9. 50 962 
9.51005 

9.51 048 
9.51 091 
95^ 134 
9.51 i7f 



45 

45 

45 

45 

45 

45 

44 

45 

45 

45 

44 

45 

44 

44 

45 

44 

44 

44 

4^ 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

4T 

41- 

43 

44 

44 

44 

43 

44 

43 

44 

43 

43 

44 

43 

43 

43 

43 

43 

43 

43 

43 

.43 

43 

43 

43 

43 

43 

43 

43 

43 



51 46O 
51 421 
51 376 
51 330 

51 285 



51 240 

51 195 
51 151 
51 106 
51 061 



51 016 
50971 

50926 
50 882 

5083? 



50792 
50748 

50703 
50659 
50614 



50570 

50525 
50481 

50437 

5039 2_ 

50 348" 
50304 
50 260 
50 216 

50 172 



50 128 
50083 
50039 
49996 

49_952^ 
49 908 
49 864 
49 826 

49 776 

49 733 



IjOU'. Cos. 



49 689 

49 645 
49602 

49558 
49 5 U 



49 47 1 
49427 
49384 
49340 
49297 



49254 
49 216 
49167 

49 124 
49 081 



49038 
48994 
48 951 

48908 
48 865 

48 822 



9.98059 
9.98 056 
9. 98 052 
9.98 048 
9.98044 



9. 98 040 
9.98036 
9.98032 
9.98:028 
9.98024 



9.98 02 1 
9.98 017 
9.98013 
9.98 009 
9.98005 



9.98 001 
9-97 997 

9-97 993 
9.97989 
9.97985 



9.97981 
9.97 977 

9-97 973 
9.97969 

9-97966- 
9.97962 
9.97958 

9-97 954 
9.97950 
9.97946 



9-97 942 
9-97938 
9-97 934 
997930 
9.97926 



ivOi;. Cot. I ('. <i. I liOi;. Tan. 



9.97922 
9.97918 
9.97914 
9.97910 
9.97906 



9.97902 
9-97898 
9.97894 

9-97 890 
9.97886 



9.97 881 
9.97 Syf 

9-97873 
9-97 869 
9.97865 



9.97 861 

9.97857 

9-97853 

9-97849 
9.97845 



9.97841 
9-97837 

9-97833 
9.97829 
9.97824 



9.97 820 



LoL'. Sin. 



3 
4 
4 
4 
3 
4 
4 
4 
4 

3 
4 
4 
4 
4 
4 

4 
4 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 



<»0 

59 
58 

57 

55 
54 
53 
52 

50 

49 
48 

47 
46 



45 
44 
43 
42 
41 
40" 
39 
38 
37 
_3l 
35 
34 

32 

JL 
30 

29 
28 

27 
26, 



r. 1'. 



24. 



21 



20 

19 
18 

17 
16 



6 


4.5 


4.5 


4- 


7 


5.3 


5.2 


5- 


8 


6.6 


6.0 


5. 


9 


6.8 


6.f 


6. 


10 


7.6 


7-5 


7- 


20 


15. 1 


15.0 


14. 


30 


22.7 


22.5 


22. 


40 


30.3 


30.0 


29. 


50 


37-9 


37-5 


37- 



4S 45 44 44 

4 4-4 
2 5.1 

9 5-8 
7, 6.6 
4' 7-3 
8 '4-6 
2 22.0 

6 29-3 
1:36.6 



43 

4-3 
5-0 
5-^ 
6.4 
7.1 

14.3 
21.5 

28.6 

35-8 





43 


6 


4-3 


7 


5-' 


8 


S-8 


9 


6.5 


10 


7.2 


20 


14.5 


30 


21.^ 


40 


29.0 


50 


36.2 





4 


4 


6 


0.4 


0.4 


7 


0.5 


0.4 


8 


0.6 


0.5 


9 


0.7 


0.6 


10 


0.7 


0.6 


20 


1.5 


1-3 


30 


2. 2 


2.0 


40 


3.0 


2.6 


50 


3.7 


3-3 



3 

0.3 
0.4 
0.4 
0.5 
0.6 
i.T 
i.f 

2.3 
2.9 



1'. I'. 





41 


41 


40 


40 


6 


4.1 


4.1 


4.0 


4.0 


7 


4-8 


4.8 


4-7 


4-«6 


8 


5-5 


5.4 


5-4 


•5-3 


9 


6.2 


6.T 


6.1 


^.0 


10 


6.9 


6.8 


6.^ 


•6.6 


20 


13-8 


13-6 


13-5 


13-3 


30 


20.^ 


20.5 


20.2 


20.0 


40 


27-6 


27-3 


27.0 


26.6 


50 


34-6 


34-1 


33-7 


33-3 





39 


39 


38 


6 


3-9 


3-9 


3-8 


7 
8 


4.6 
5-2 


4-5 
5-2 


4-5 . 
5-1 


9 

lO" 


5-9 
6.6 


5 8 
6.5 


5-« : 
6.4 


20 


13.1 


13.0. 


12.8 


30 
40 


19.7 

26.3 


195 
26.0 


19.2 
25-6 


50 


32-9 


32.5 


32.1 



4 -w 



3^-5 



TABLE VIL 



Log. Sin. 



-LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 

18° 



10 

II 

12 

! l6 

17 
i8 

19 



9.48 998 

9-49 0-37 
9.49076 

9.49 114 

9-49 153 



9.49192 
9.49231 
9.49269 
949308 

9-49 346 



20 

21 

22 

23 
_24_ 

25 
26 

27 
28 

29 

80 

31 
32 

33 

34 



35 
36 
37 
38 

il 
40 

41 
42 
43 
44 

45 
46 

47 
48 

49 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 
00 



949385 
9-49423 
9.49462 
9.49 500 

9-49 539 



9 49 577 
9.49615 

9-49653 
9.49692 

9-49730 



9.49760 
9.49805 

9.49 844 
9.49882 
949920 

9 49 958 
9 49 996 
9.50034 
9.50072 

9.50 no 



9.50147 
9.50185 
9.50223 
9.50 265 
9. 50 293 



9-50336 

9-50373 
9. 50 41 1 

9- 50 448 

9. ;o 486 

9-5o'52j 
9.50561 
9.50598 
9.50635 
9. 50 672 



9.50710 
9-50747 
9-50784 
9.50 821 

9.50858 



9.50895 
9.50932 

9. 50 969 

9.51 006 
9-51043 



9. 5 1 080 
9.51 117 
9.51 154 
9.51 190 
9.51 22^ 



9.51 264 

Loff. Cos. 



(1. I liOe:. Tan. 



39 

39 

38 

39 

38 

39 

38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

j>^ 

3^ 

3^ 

38 

38 

38 

38 

3^ 

38 

38 

37 

38 

38 

37 

38 

37 

37 

38 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
36 
37 
36 
37 

36 
(I. 



9.f,i 177 
9.51 226 
9.51 263 

9-51 306 
9-51 349 
9.51 392 

9-51435 
9-51477 
9.51 520 
9.51 563 



9.51 605 
9.51 648 
9.51 691 

9-51 733 
9.51 776 



9.51 818 
9.51 861 
9.51 903 
9.51 946 
9.51 988 



9.52 030 
9.52073 
9.52 115 

9.52 157 
9 52 199 



9.52241 
9.52284 
9.52 326 
9.52 368 
9.52410 



9.52452 
9.52494 
9.52536 
9.52578 
9.52 619 



9.52 661 
9.52703 

9-52745 
9.52787 

9-52828 



9.52870 
9.52 912 

9 52953 
9.52995 

9 53036 



9.53078 

9-53 119 
9.53 161 
9.53 202 
9-53244 



9.53285 

9-53 326 
9- 53 368 

9-53409 
9-53450 



9-53491 
9-53 533 
9-53 574 
9-53615 
9.53656 



9-53697 



C. (I. 



43 

43 
43 
43 
42 

43 

42 
43 
42 

42 

43 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 

42 
42 

42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
41 
42 
42 

41 
42 

41 

41 
42 

41 
41 
41 
41 

41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 



Los?. Cot. 



0.48822 
0.48 779 

0.48 736 
0.48 693 
0.48 656 



0.48 608 
0.48 565 
0.48 522 
0.48 479 
0-48437 
0.48 394 
0.48 351 
0.48 309 
0.48 265 
0.48 224 



0.48 181 
0.48 139 
0.48 096 
0.48 054 
0.48 012 



0.47 969 
0.47 927 
0.47 885 
0.47 842 
0.47 806 



0-47 758 
0.47 716 
0.47 674 
0,47 632 
0.47 590 



0.47 548 
0.47 506 
0.47 464 
0.47 422 
0.47 386 



0-47 338 
0.47 296 
0.47 255 
0.47 213 
0.47 1 71 



0.47 130 
0.47 088 

0.47 046 
0.47 005 
0.46 963 



0.46 922 
0.46 886 
0.46 839 

0.4679? 
0.46 756 



0.46 714 
0.46673 
0.46 632 
0.46 591 
0.46 549 



Lot?. Cot. I c. d. 



0.46 5O8 
0.46 467 
0.46426 
0.46 385 

0-46 344 
0-46 303 

Log^. Tan. 



Los. Cos. 



9.97 826 

9-97816 
9.97 812 

9.97 808 
9.97 804 



9.97 800 
9-97796 
9-97792 
9.97787 

9-97783 



9-97 779 
9-97 775 
9.97771 
9.97 767 
9.97 763 



9-97 758 
9-97 754 
9-97750 
9-97746 
9-97742 



9-97 73? 
9-97 733 
9.97729 

9.97725 
9-97721 



9-97 716 
9.97712 

9-97 708 
9.97704 
9-97 700 



9.97695 
9.97691 
9-97687 
9.97 683 
9-97678 



9.97674 
9.97670 
9.97 666 
9.97 661 
9.9765^ 



9-97653 
9.97649 
9.97644 
9.97646 
9.97636 



9.97632 
9-97 62f 
9.97623 
9.97619 
9.97614 



9.97 616 
9.97 606 
9.97 601 
9-97 59? 
9-97 593 



9-97 588 
9.97584 
9.97580 
9.97 575 
9-97 571 
9-97 567 

Log. Sin. 



4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

"dT 



GO 

59 
58 
57 
56 



45 
44 

43 

42 

41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 

24 

23 

22 

21 

19 
18 

17 
16 



p. p. 



15 

14 

13 
12 
II 



10 

9 
8 

7 
6 





43 


42 


6 


4-3 


4.2 


7 
8 

9 


5.0 

5.? 
6.4 


4.9 

5-6 

6.4 


10 


7-1 


7-1 


20 


14-3 


14.1 


30 


21. s 


21.2 


40 


28.6 


28.3 


50 


35-8 


354 



42 

4.2 

4-9 
5.6 

6.3 
7.0 
14.0 
21.0 
28.0 
35-0 





41 


6 


4-1 


7 


4-8 


8 


5-5 


9 


6.2 


10 


6.9 


20 


13-8 


30 


20.7 


40 


27-6 


50 


34-6 



41 

4.1 

4-8 

5-4 
6.1 

6.8 

13-6 
20.5 

27-3 
34-1 





39 


38 


6 


3-9 


3-8 


7 


4-5 


4-5 


8 


5.2 


5-1 


9 


58 


5.8 


10 


6.5 


6.4 


20 


13.0 


12.8 


30 


195 


19.2 


40 


26.0- 


25-6 


50 


32.5 


32.1 



6 

7 
8 

9 
10 

20 

30 
40 

50 



31 

3.? 
4.4 
5.0 

5.6 
6.2 
12.5 
18.^ 
25.0 
31.2 



37 

3-7 

4.3 
4.9 

5-5 
6.1 
12.3 
18.5 
24.6 
30.8 



38 

3-8 
4.4 
5.6 
5-7 
6.3 
12.6 
19.0 

25.3 
31-6 



36 

3-6 

4.2 

4-8 

5-5 

6.1 

12. T 

18.2 

24-3 
30.4 



6 

7 
8 

9 
10 

20 

30 

40 

50 



4 

0.4 

0.5 

0.6 

0.7 

o.? 

1-5 
2.2 

3-0 
3-? 



4 

04 
0.4 

o.S 
0.6 

0.6 

1-3 
2.0 

2.6 
3-3 



P. p. 



366 



TABLE VII. -LOGARITHMIC SINES. COSINES. TANGENTS, AND COTANGENTS 

19 



10 

II 

12 

13 

1± 

15 

i6 

17 
i8 

19 



20 

2; 

22 



25 

26 
27 
28 
29 



30 

31 
32 

33 
34 



35 

36 
37 
38 
39 



40 

41 

42 

43 
4£ 

45 
46 

47 
^8 

49 



r>o 

51 
52 
53 
54 



55 

56 

57 
58 
59 

(>0 



Lo:;. Sill. 



(]. 



9- 
9- 
9' 
9. 
9: 
9- 
9 
9 
9 
9 

9 
9 
9 
9 
9 



264 
301 
337 
374 
416 



447 
483 
520 

556 
593 



629 
665 
702 
738 
774 



816 

847 
883 
919 
955 



51 991 

52 027 
52 063 
52099 
52 135 



52 170 
52206 
52242 
52278 
52314 



52349 
52385 
52421 

52456 
52492 



5252? 
52563 
52598 
52634 

52 669 



52704 
52740 

52775 
52 810 

52846 



52881 
52 916 
52951 

52986 
53021 



53056 
53091 
53 126 
5316T 

53 196 



53231 
53 266 
53301 
53 335 
53370 

9-53405 

Log. Cos. 



37 
36 
36 
36 
36 
36 
36 
36 
36 

36 
36 

36 
36 

36 
36 
36 
36 
36 
36 
36 
36 
36 
36 

36 



36 
3l 
36 

35 
3l 
36 
35 
35 
35 
35 

JD 

:>:> 
35 
35 
35 
35 
35 
35 
35 
35 
3:) 
35 
35 
35 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 



og. Tail. I c. (I. I Log. Cot. 



53697 
53738 

53 779 
53 S20 
53861 



53902 

53 943 
53983 
54024 
54065 



54 106 

54147 
54187 
54228 
54269 



54309 
54350 
54390 
54431 
54471 



54512 
54552 
54 593 
54633 
54673 



54714 
54 754 
54 794 
54834 
54874 



54915 
54 955 
54 995 
55035 
55075 



55115 

55 155 
55 195 
55235 

55275 



55315 
55 355 
55 394 
55 434 
55 474 



55 514 
55 553 
55 593 
55633 
55672 



55712 
55751 
55791 
55831 
55870 



55909 

55 949 
55988 
56028 
5606^ 

56 106 

LOK. Cot. 



41 
41 
41 
41 
41 
41 
40 
41 
41 
46 

41 
40 
40 
41 
40 
40 
40 
40 
40 

40 
40 
40 
40 
40 
40 
40 
46 
40 
40 

40 
40 
40 
40 
40 

40 

39 
40 
40 
40 
40 
40 
39 
40 
39 
40 

39 
40 

39 
39 
39 
39 
40 

39 
39 
39 
39 
39 
39 
39 

39 



0.46 303 
0.46 262 
0.46 221 
0.46 180 
0.46 139 



0.46 098 
0.46057 
0.46 015 

0.45975 
0.45 934 



0.45 894 

0.45 853 
0.45 812 
0.45 772 

0.45731 



0.45 690 
0.45 650 
0.45 609 

0.45 569 

o 45 528 



0.45 488 
0.45 447 
0.45 407 
0.45 367 
0.45 326 



0.45 286 
0.45 246 
0.45 205 

0.45 165 
0.45 125 



0.45 085 

0.45045 
0.45 005 

0.44965 
0.44925 



0.44884 
0.44845 
0.44 805 
0.44 765 
0.44725 



0.44 685 
0.44645 
0.44 60 5 

0.44 565 
0.44 526 



0.44486 

0.44 446 
0.44406 
0.44 367 
0.44327 



0,44 288 
0.44 248 
0.44 208 
0.44 169 
0.44 129 



0.44090 
0.44 051 
0.44 01 T 
0.43 972 
0.43 932 
0.43893 

Loir. Tan. 



Loir. Cos. 



9.97 567 
9.97 562 

9.97 558 
9-97 554 
9-97 549 



9-97 545 
9-97 541 
9-97 536 
9-97 532 
9.97 527 



9-97 523 
9.97 519 
9.97514 
9.97 510 
9.97 505 



9.97 501 
9-97 497 
9-97492 
9.97488 

997483 



9-97 479 
9-97 475 
9.97476 
9.97466 
9-97461 



9-97 457 
9.97452 
9.97448 
9-97 443 
9-97 439 



9-97 434 
9-97430 
9-97425 
9.97421 

9-97 416 



9-97412 
9.97407 

9-97403 
9-97 398 
9-97 394 



9-97 389 
9-97385 
9-97 380 
9-97376 
9-97371 



9-97 367 
9-97 362 
9-97358 
9-97 353 
9-97 349 



9-97 344 
9-97 340 
9-97 335 
9-97330 
9-97 326 



9-97321 
9-97317 
9.97312 

9-97 308 
9-97303 
9 97 298 

Loif. Sin. 



<l. 



4 

4 
4 
4 
4 
4 
4 
4 
•4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
5 
4 
4 
4 
4 
4 
4 
5 
4 



00 

59 
58 
57 
56 



55 
54 
53 
52 
_5i 

r>o 

49 
48 

47 

^6^ 

45 
44 

43 
42 

41 



24 



21 



20 

'9 

18 

17 
16 



15 
14 

13 
12 

1 1 

1(7 

9 
8 

7 
6 

5 
4 
3 



r. I'. 





41 


40 


40 


6 


41 


4.6 4.0 1 


7 


4.8 


4-7 


46 


8 


5-4 


5-4 


5-3 


9 


6.1 


6.1 


6.0 


10 


6.8 


6.f 


6.6 


20 


■3-6 


'3-5 


13-3 


30 


20.5 


20.2 


20.0 


40 


27.3 


27.0 


26.6 


50 


34-1 


33-i^ 


33-3 



39 



6 

7 
8 

9 
10 

20 

30 
40 

50 



3.9 


3- 


4.6 


4- 


5-2 


5- 


5.9 


5- 


6.6 


6. 


13-1 


13- 


19-7 


19- 


26.3 


26. 


32.9 


32- 



39 

9 





37 


36 


6 


3-7 


3-6' 


7 


4-3 


4-2 


8 


4.9 


4.8 


9 


5-5 


5-5 


10 


6.T 


6.1 


20 


12.3 


12. 1 


30 


18.5 


18.2 


40 


24-6 


24-3 


50 


30.8 


30.4' 



36 

3-6 
4.2 
4-8 

5-4 
6.0 
12.0 
18.0 
24.0 
^o.o 



35 35 34 



/ 
8 

9 
10 

20 
30 
40 
50 



J 5 


3-5 


3- 


4 


I 


4.1 


4- 


4 


7 


4-6 


4- 


5 


3 


5.2 


5- 


5 


9 


5-8 


5- 


II 


8 


II. 6 


1 1. 


17 


7 


17-5 


17- 


23 


6 


23-3 


23- 


29 


6 


29.1 


28. 



6 

7 
8 

9 
10 

20 

30 
40 

50 



5 

0.5 
0.6 

0.6 
o.'7 
0.8 
1-6 
2-5 
3-3 
4.1 



4 

0.4 

0.5 
0.6 
0.7 
o.f 
1.5 



3-f 



4 

0.4 
0.4 

0.5 
0.6 

0.6 
1-3 
2.0 

2.6 
3-3 



O 



3(>1 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 



10 

II 

12 

13 
14 

15 
16 

17 
18 

19 

20 

21 
22 
23 

-7 1 

— -T 

25 
26 
27 
28 

30 

31 

32 
33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 
50 

51 
52 
53 
54 

55 
56 
i 57 
58 
59 
60 



Lost. Mil 



53 405 
53440 

53 474 
53509 
53 544 



53 578 
53613 
53647 
53682 

53716 



53750 
53785 
53819 
53854 
53888 



53922 
53 956 

53990 
54025 

54059 



54093 
54127 

5416T 

9 54195 
9 54229 



9.54263 
9.54297 

9-54331 
9-54365 
9-54 398 



9-54 432 
9- 54 466 
9- 54 500 
9-54 534 
9-54567 



9.54601 

9-54634 
9.54668 
9.54702 

9-54 73? 



9.54769 
9. 54 802 
9-54836 
9.54869 
9 54 902 



9-54936 
9.54969 
9.55002 
9- 55 036 
9-55069 



9-55 102 
9-55 135 
9-55 168 
9.55 202 

9-55 235 



9-55 



268 



9.55301 
9-55 334 
9-55367 
9-55400 

9-55 433 

Loff. Cos. 



d. 



35 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
3-4 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
34 
33 
34 
34 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 



Los. Tail. 



9.56 IO6 
9.56 146 
9.56 185 
9.56 224 
9.56263 



9-56303 
9.56342 

9-56381 
9.56420 
9.56459 



9.56498 
9-56537 
9-56576 
9.56615 
9.56654 



56693 

56732 
56771 
56810 
56848 



9.56 887 
9.56 926 
9.56965 
9-57003 
9-57042 



9.57081 
9.57 119 
9.57 158 

9-57 196 
9-57235 



9-57274 
9.57 312 

9-57350 
9-57389 
9-57427 



9.57466 
9-57 504 
9-57 542 
9.57581 
9.57619 



9-57657 
9.57696 

9-57 734 
9-57772 
9.57810 



9-37848 
9-57886 
9.57925 

9-57963 
9.58 001 



9.58039 
9-58077 
9.58 115 

9-58153 
9.58 196 



9.58 228 

9.58266 

9-58304 
9.58342 

9-58380 

9-58417 
Lost. Cot. 



c, (1. 



39 
39 

39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
38 
39 
39 
39 
39 
38 
39 
38 
39 
38 
38 
39 
38 
38 
38 
38 
39 
38 
38- 
38 
38 
38 
38 
38 
38 
38 
38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

37- 

38 

38 

38 

37 

38 

37 



Log. Cot. 



0.43 893 
0.43 854 

0.43815 
0.43775 

0.43 736 



0.43 697 
0.43658 
0.43619 
0.43 580 
0.43 540 



0.43 501 
0.43 462 
0.43423 
0.43 384 
0.43 346 



0.43 307 

0,43 268 

0.43 229 
0.43 190 

0.43 151 



0.43 1 12 
0.43074 

0.4303 s 
0.42 996 
0.42 958 



0.42 919 
0.42 880 
0.42 842 
0.42 803 

0.42 765 



0.42 726 
0.42 687 
0.42 649 
0.42 611 

0.42 572 



0.42 534 
0.42 495 
0.42457 
0.42 419 
0.42 380 



0.42 342 
0.42 304 
0.42 266 
0.42 227 
0.42 189 



0.42 151 

0.42 113 
0.42075 
0.42037 
0.41 999 



0.41 961 
0.41 923 
0.41 885 
0.41 847 
0.41 809 



0.41 771 

0.41 733 
0.41 6gl 
0.41 658 
0.41 620 
0.41 582 

c. (I. I Loi?. Tan. 



Loir. Cos. 



9.97 298 
9.97 294 
9.97 289 
9-97 285 
9.97 280 



9-97275 
9.97271 

9.97 266 
9.97 261 
9.97257 



9.97 252 
9.97 248 

9-97 243 
9-97238 
9-97 234 

9-97 229 

9.97 224 
9.97 220 
9.97215 
9.97 210 



9.97 206 
9.97 201 
9-97 



9-97 
9-97 



9-97 
9-97 
9-97 
9-97 
9-97 



9-97 
9-97 
9-97 
9-97 
9-97 



9-97 
9-97 
9-97 
9-97 
9-97 



9-97 

9-97 

9-97 

9.97097 

9.97092 



59 
54 
49 
44 
40 



35 
30 

25 
21 

16 



II 
06 



02 



9.97087 
9.97 082 
9.97078 
9.97073 
9.97068 



9.97063 

9-97058 
9.97054 
9.97049 
9.97044 



9-97039 
9-97034 
9.97029 
9-97025 
9.97 020 

9-97015 

Lost. Sin. 



4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 

5 
4 
4 
5 
4 
4 
5 
4 
5 
4 

4 

5 
4 
5 
4 
4 

5 
4 
5 
4 

5 
4 
5 
4 
5 
4 
5 
4 
5 
5 
4 
5 
4 
5 
4 

5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 



60 

59 
58 
57 
56 



55 
54 
53 
52 
^i 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
V_ 

30 

29 
28 
27 
26 



25 
24 
23 
22 
21 



20 

19 

18 

17 
16 



15 
14 

13 
12 

II 



10 

9 

8 

7 
6 



p. P. 





39 


3S 


6 

7 
8 


3.9 
4.6 
5-2 


3- 

4. 
5- 


9 
10 


5-9 
6.6 


5- 
6. 


20 


13-1 


13- 


30 
40 


19.7 
26.3 


19. 

26. 


50 


32.9 


32- 



6 

7 
8 

9 
10 
20 

30 
40 

50 



38 

3-8 
4-5 
5-1 
5-8 
6.4 
12.8 
19.2 

25-6 
32,1 



38 

3-8 
4.4 
5.6 

5-7 

6.3 

12.6 

19.0 

25-3 
31-6 



27 

3-7 

4 

5 



5 
6 

12 

18 

25 

31 





35 


34 


6 


3-5 


3-4 


7 


4.1 


4.0 


8 


4-6 


4.6 


9 


5-2 


5.2 


10 


5-8 


5-7 


20 


II. 6 


11-5 


30 


17-5 


17.2 


40 


23-3 


23.0 


50 


29.1 


28.? 



34 

3-4 
3-9 
4-5 
5-1 
5-6 
II-3 
17.0 

22.6 
28.3 



6 


3-3 


3- 


7 


3-9 


3. 


8 


4-4 


4- 


9 


5-0 


4- 


10 


5.6 


5- 


20 


II. I 


II. 


30 


16.7 


16. 


40 


22.3 


22. 


50 


27.9 


27- 



23 22 



4 
0.4 

0.5 
0.6 

0.7 
0.7 

1-5 
2.2 

3-0 

3-7 





5 


6 


0.5 


7 


0.6, 


8 


0.6! 


9 


0.7 


10 


0.8 


20 


1-6 


30 

40 


2-5 

3-3 


50 


4.1 



p. p. 



G9 



368 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

2\ 



10 

II 

12 

14 



15 
16 

18 
19 



20 

21 
22 

23 
24 



26 

27 
28 
29 



80 

31 
32 
33 
34 



J3 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



Locr. Sill. 



9-55 433 
9.55466 

9-55 498 
9-55 531 
9-55 564 



9-55 597 
9.55630 
9.55662 
9.5569^ 
9.55728 



9.55760 

9-55 793 
9.55826 

9-55^58 
9.55891 



(1. 



9-55923 
9-55956 

9-55 9S8 
9.56 020 
9.56053 



'9. 56085 
9.56 118 
9.56 150 
9.56 182 
9.56214 



9.56247 
9.56279 
9.56311 
9-56343 
9-56375 



9- 56 407 
9-56439 
9.56471 
9.56503 

9-56533 



9.56567 

9-56599 
9.56631 
9.56663 
9.56695 



9.56727 

9-56758 
9. 56 790 
9.56 822 
9.56854 



50 

51 
52. 
53 
54 



55 
56 
57 
58 
59 



60 



9.56885 
9.56917 

9- 56 949 
9 56 980 
9.57012 



9- 57 043 
9.57075 

9-57 106 
9-57 138 
9-57 169 



9.57 201 
9.57232 
9-57263 
9.57295 
9-57 326 



9-57 357 



lj(»ff. Cos. 



33 
32 
jj 
33 
32 

33 
32 
33 
32 
32 
32 
33 
32 
32 
32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
31 
32 

32 
31 
32 
31 
32 

3f 
31 
32 

31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 



Lost. Tan. 0. d. I Loe. Cot. 



9.58417 
9.58455 

9- 58 493 
9-58531 
9.58568 



9.58 606 
9.58644 
9.58681 
9.58719 
9-58756 



9.58794 
9-58831 
9.58869 

9-58906 
9 58 944 



9.58 981 
9.59019 
9.59056 

9- 59 093 
9-59 131 



9.59 168 
9-59205 
9.59242 
9.59 280 
9.59317 



9-5935 + 
9-59391 
9.59428 
9.59465 
9.59502 



9.59540 

9-59 577 
9.59614 
9.59651 
9.59688 



9-59724 
9-59761 
9-59 798 

9-59833 
9.59872 



9. 59 909 

9- 59 946 
9.59982 
9.60019 
9 60056 



9.60093 
9.60 129 
9.60 165 
9.60 203 
9.60239 



d. 



9.60 276 
9.60 312 

9-60349 
9.60 386 
9.60422 



9-60459 
9.60495 
9.60 531 
9.60 568 
9.60 604 



9. 60 64 1 



38 

3? 
38 
37 

3f 
38 
37 
3f 
37 
37 
37 
3? 
37 
3? 
3? 
3l 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

36 

37 

37 

37 

36 

37 

37 

36 

37 

36 

37 

36 

37 

36 

36 

36 

36 

37 

36 

36 

36 

36 

36 

36 

36 

36 



0.41 582 
0.41 544 
0.41 507 
0.41 469 
0.41 43? 



0.41 394 
0.41 356 
0.41 318 
0.41 281 
0.41 243 



0.41 206 
0.41 163 
0.41 131 
0.41 093 
0.41 056 



0.41 oig 
0.40 981 
0.40 944 
0.40 906 
0.40 869 



0.40 832 
0.40 794 
0.40757 
0.40 720 
0.40 683 



0.40 646 
0.40 608 
0.40 571 

0.40 534 
0.40 497 



0.40 460 
0.40423 
0.40 386 
0.40 349 
0.40 312 



l,nir. ("OS. 



0.40 275 
0.40 238 
0.40 201 
0.40 164 
0.40 128 



0.40 091 
0.40 054 
0.40017 
0.39 980 
0.39944 



o. 39 907 
0.39876 

0-39833 
0.39797 
0.39766 



0.39724 
0.3968^ 
0.39650 
0.39614 
0.3957? 



0.39541 
0.39 504 
0.39468 
0.39432 
0.39395 



9.97015 
9.97 016 
9.97005 
9.97 000 
9.96995 



9.96991 
9.96 986 
9.96 981 
9.96976 
9-96971 



9-96966 
9.96 961 

9-96956 
9-96952 
9-96947 
9-96942 
9-96937 
9.96932 

9-96927 
9.96 922 



9.96917 
9.96 912 

9-96907 
9. 96 902 
9-9689? 



9.96 892 
9.9688? 
9.96882 
9.96877 
9.96873 



9.96868 
9.96 863 
9.96858 
9.96853 
9.96848 



9.96843 
9.96838 

9-96833 
9.96828 
9.96823 



9.96 818 
9.96 813 
9.96808 
9.96 802 
9.96797 



9-96792 
9.96787 
9.96 782 

9-9677? 
9.96772 



9-96 767 
9.96 762 

9-9675? 
9.96752 

9.96747 



o- 39 359 



Cot. c. d. : liOtr. 'In 11. 



9 96 742 
9-96737 
9.96732 
9.96727 
9.96 721 



9.96 7 16 



4 
5 
5 
5 
4 
5 
5 
5 
4 

5 
5 
5 
4 
5 
5 
5 
5 
5 
4 

5 
5 
5 
5 
5 
5 
5 
5 
5 
4 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
I 
5 
5 
5 
5 
5 
5 
5 



45 
44 
43 
42 
41 



40 

39 
38 
37 

35 
34 
33 
32 
31 



I'. I'. 



30 

29 
28 

27 
26 



l,ou'. sin. 



25 
24 

23 
22 
21 



20 

'9 
18 

17 
16 





38 


37 


37 


6 


3-8 


3-? 


3-7 , 


7 


4.4 


4-4 


4 


3 > 


8 


S.o 


5.0 


4 


9 


9 


S-7 


5.6 


5 


^ 


10 


6. .3 


6.2 


6 


I 


20 


12.6 


12.5 


12 


3 


30 


19.0 


18.? 


18 


5 


40 


25-3 


25.0 


24 


6 


50 


31-6 


31.2 


30 


8 





36 


36 


6 


3-6 


3-6 


7 


4.2 


4-2 


8 


4.8 


4.8 


9 


5-5 


5-4 


10 


6.1 


6.0 


20 


12. T 


12.0 


30 


18.2 


18.0 


40 


24-3 


24.0 


50 


30.4 


30.0 



33 32 32 



6 

7 
8 

9 
10 
20 

30 
40 
50 



3-3 


3-2 


3-2 


3 


8 


3 


8 


3-? 


4 


4 


4 


3 


4.2 


4 


9 


4 


9 


4.8 


5 


5 


5 


4 


5-3 


II 





10 


8 


10.6 


16 


5 


16 


2 


16.0 


22 





21 


6 


21.3 


27 


5 


27 


I 


26.6 



6 

7 
8 

9 
10 

20 
30 
40 
50 



6 

7 
8 

9 
10 
20 
30 
40 
50 



31 31 

3-1 
3-6 
4.1 

4.6 

5-1 
10.3 

15-5 
20.6 
25 8 



3 


T 


3 


7 


4 


2 


4 
5 


7 
2 


10 


5 


15 


? 


21 





26 


T 



0.5 
0.6 


0.5 
0.6 


0.? 


0.6 


0.8 


0.7 


0.9 


0.8 


1-8 
2.? 
3-6 
4-6 


1-6 

2-5 

3-3 
4.1 



4 
0.4 

0.5 
0.6 

0.7 

0.7 

'•5 
"^ 2 

30 

3-? 



1'. I" 



G8^ 



3'-^y 



TABLE VII.-LOGARITHMIC SINES, COSINES. TANGENTS. AND COTANGENT^ 

23° 



Log. Siu. 



9-57 35^ 

9-57389 
9.57420 

9-57 451 
9-57482 



9-57513 
9-57 544 
9-57576 
9.57607 
9.57638 



10 

II 
12 

13 
14 



15 
16 

17 
18 

19 



20 

21 

22 

23 

24 



9.57669 
9.57700 

9-57731 
9.57762 
9.57792 



9-57823 
9-57854 
9.5788$ 
9-57916 
9-57 947 



25 
26 

27 
28 

29 



30 

31 
32 
33 
-31 
35 
36 
37 
38 
39 



9-57 977 
9-58008 
9.58039 
9.58070 

9.58 TOO 



9-58 131 
9.58 162 
9.58192 
9.58223 
9-58253 



9.58284 
9-58314 
9-58345 
9-5837$ 
9.58 406 



9.58436 
9.58465 
9.58497 
9.58527 
9-58557 



40 9.58587 

41 9.58618 

42 9.58648 

43 9-58678 
9-58708 




50 

51 

52 
53 
54 




9-58738 
9.58769 

9.58799 
9.58 829 

9.58859 



9.58889 
9.58919 
9- 58 949 

9-58979 
9.59009 



9-59038 
9- 59 068 
9- 59 098 
9.59128 

9-59158 



9-59 188 

Log. Cos. 



31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

30 

31 

31 

31 

30 

31 

30 

31 

30 

31 

30 

30 

31 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 
30 
29 

30 
30 
29 
30 
30 



d. Log. Tail. c. <1 



9.60 641 
9.60677 
9.60713 
9.60 750 
9.60 785 



9.60 822 
9.60 859 
9.60 895 
9-60931 
9.60 967 



9.61 003 
9.61 039 
9.61 076 
9.61 112 
9.61 148 

9.61 184 
9.61 220 
9.61 256 
9.61 292 
9.61 328 

9-6i 364 
9.61 400 
9.61 436 
9.61 472 
9.61 507 



9.61 543 

9-61 579 
9.61 615 
9.61 651 
9.61 685 



9.61 722 
9.61 758 
9.61 794 
9.61 829 
9 61 865 



9.61 901 

9.61 936 

9.61 972 

9. 62 007 
9-62043 

9.62078 
9.62 114 
9.62 149 
9.62 185 
9.62 220 

9.62 256 
9.62 291 
9.62 327 
9.62 362 
9-62 39^ 

9-62433 
9.62468 
9.62 503 
9-62 539 
9-62 574 
9.62 609 
9.62 644 
9.62 679 
9.62 715 
9.62 750 



9.62 785 



36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 

36 
36 
36 
36 

36 
36 
36 
36 
3! 
36 
36 
3§ 
36 
35 
36 

3l 

36 

35 

35 

36 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 



Log. Cot. 



0-39359 
0.39322 

0.39285 
0.39250 
0.39213 



0.39 177 
0.39 141 
0.39 105 
0.39069 
0.39032 



0.38995 
o. 38 966 

0.38924 
0.38888 

0.38852 



0.38816 
0.38 780 
0.38744 
0.38 708 

0.38 672 



0.38636 

0.38 600 

0.38 564 
0.38 528 

0.38492 



0-38456 

0.38 420 

0.38385 

0.38349 
0.38 313 



0.3827^ 

0.38 242 
0.38 206 
0.38 170 

0.38 135 



0.38099 

0.38 063 

0.38028 
0.37 992 

0.37 957 



0.37 921 
0.37 S86 
0.37 850 
0.37815 

0.37779 



0-37 744 
0.37 708 
0.37673 
0.3763^ 
0.37 602 



0-37 567 

0.37 531 

0-37 496 
0.37461 

0.37426 



Log. Cot. led. 



0.37390 

0.37355 
0.37326 

0.37 28 5 
0.37250 



Log. Cos. 



o- 37 215 

Log. Tan. 



9.96715 
9.96 711 
9.96705 
9.96 701 
9.96 696 



9.96 691 
9.96686 
9.96681 
9-9667$ 
9.96 676 



9.96665 
9.96660 
9.96655 
9.96 650 
9-96644 



9.96639 
9.96634 
9.96 629 
9.96 624 
9.96 619 



9.96 613 
9-96608 
9. 96 603 
9.96598 
9.96593 



9-96 587 
9.96 582 
9.96577 
9-96572 
9-96 567 



9.96 561 

9-96556 
9.96551 

9.96546 

9-96 540 



9-96 535 
9-96 530 
9.96525 
9.96519 
9.96514 



9.96 509 

9-96 503 
9.96498 

9-96493 
9.96488 



9.96482 

9-96477 
9.96472 
9.96465 
9.96461 



9.96456 

9-96450 

9-96445 
9.96440 

9-96434 



9-96429 

9-96424 

9-96418 
9.96413 

9. 96 408 
9.96 402 

Log. 8111. 



5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 



d. 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 

43 
42 

41 



40 

39 
38 
37 
36 



35 
34 
33 
32 

_3i 
30 

29 
28 
27 
26 



25 
24 

23 
22 
21 



20 

19 
18 

17 
16 



15 

14 

13 
12 

II 

To 

9 
8 

7 
6 

5 
4 
3 
2 
I 

~0~ 



P. p. 





36 


6 


3-6 


7 


4.2 


8 


4.8 


9 


5-5 


10 


6.1 


20 


12. 1 


30 


18.2 


40 


24-3 


50 


30.4 





3S 


6 


3-5 


7 


4.1 


8 


4.? 


9 


5-3 


10 


5-9 


20 


11.8 


30 


17-7 


40 


23-6 


50 


29.6 



7 
8 

9 
10 
20 
30 
40 
50 



3i 

3-1 

3-7 
4.2 

4-7 

5.2 

10.5 

15-^ 
21.0 
26.2 





30 


30 


6 


3-0 


3-0 


7 


3-5 


3.5 


8 


4-0 


4.0 


9 


4.6 


4-5 


10 


5-1 


5.0 


20 


10. 1 


lO.O 


30 


15.2 


15.0 


40 


20.3 


20.0 


50 


25.4 


25.0 



36 

3-6 

4.2 
4.8 

5-4 
6.0 
12.0 
18.0 
24.0 
30.0 



35 

3.5 
4-1 
4-6 
5.2 

5-8 
II. 6 

17.5 
23-3 
29.1 



31 

3-1 
3-6 

4-1 

4.6 

5-1 
10.3 

15.5 
20.6 

25-8 



29 

2.9 
3-4 
3-9 
4.4 
4-9 
9-8 
14.^ 

19-6 
24.6 





B 


5 


6 


0.5 


0-5 


7 


0.5 


0.6 


8 


0.7 


0.6 


9 


0.8 


0.7 


10 


0.9 


0.8 


20 


1-8 


1-6 


30 


2.7 


2.5 


40 


3-6 


3-3 


50 


4-6 


4.1 



p. p. 



67 



370 



TAl^LE VII.— LOGARITHMIC SINES, 



COSINES, 

2:v 



TANGENTS, AND COTANGENTS. 



10 

II 

12 

13 

14 



20 

21 
22 

23 

24 



25 
26 

27 



29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 
43 
44 

45 
46 

47 
48 

49 
50 

51 

52 
53 
54 



55 
56 
57 
58 
59 
60 



Log. Sill. I d. 



15 


9-59 


16 


9-59 


17 


9-59 


18 


9.59 


19 


9.59 



9-59 
9-59 
9-59 
9-59 
9-59 



188 
21^ 
247 
277 
306 



9-59 
9-59 
9-59 
9-59 
9-59 



336 
366 

39^ 

425 
454 



9-59 
9-59 
9-59 
9-59 
9-59 



484 
513 
543 

572 
602 



631 
661 
696 
719 

749 



9-59 
9-59 
9-59 
9-59 
9-59 



778 
807 

S37 
866 
895 



9-59 
9.59 
9-59 
9.60 
9.60 



924 

953 
982 

012 

041 



9.60 
9.60 
9.60 
9.60 
9.60 



070 
099 
128 

157 
186 



9.60 
9.60 
9.60 
9.60 
9.60 



215 

244 

273 
301 

330 



9.60 
9.60 
9.60 
9.60 
9.60 



359 
388 

417 
445 
474 



9.60 
9.60 
9.60 
9.60 
9.60 



503 
532 
560 

589 
618 



9.60 
9.60 
9.60 
9.60 
9.60 



646 
675 

703 
732 
766 



9.60 
9.60 
9.60 
9.60 
9.60 



789 
817 
846 

874 
903 



9-60931 

Log. Cos. 



29 
30 
29 
29 

30 
29 

29 
29 
29 
29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 
29 
29 
28 

29 
29 

28 

29 

28 

29 

28 

29 

28 
28 

29 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
d. 



Lou. Tan. 



9.62 785 
9.62 820 
9.62855 
9.62 890 
9.62 925 



9.62 966 
9.62995 
9.63030 
9.63065 

9.63 106 



963 135 
9.63 170 
9.63 205 
9.63 240 
9.63275 



9.63310 
9-63344 

9-63379 
9.63414 

9-63449 



9.63484 
9-63 5I8 

9-63553 
9.63 588 

9.63 622 



9-63657 
9.63 692 

9-63726 
9.63761 

9-63795 



9.63830 
9.63 864 
9.63899 

9-63933 
9-63 968 



9. 64 002 
9.64037 
9.64071 
9.64 106 
9.64 140 



9.64 174 
9.64 209 
9.64 243 
9.64 277 
9.64312 



9-64 346 
9.64 380 
9.64415 
9.64449 
9-64483 



9-6451? 
9.64551 

9.64585 
9.64620 
9.64654 



c. d. 



9.64688 
9.64 722 
9.64756 
9.64790 
9.64 824 

9.64'85§" 

Loir. Cot. led. 



35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
34 
35 
35 
3I 
35 
35 
34 

35 
34 
34 
35 
3-+ 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
3^ 
34 
34 
34 
34 

34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 



Log. Cot. 



0.37 215 
0.37 179 
0.37 144 
0.37 109 
0.37074 



0.37039 
0.37004 
o. 36 969 

0.36934 
0.36899 



o. 36 864 
0.36 829 

0.36794 

0.36 760 

0.36725 



0.36 690 

0.36655 

0.36 626 

0.36585 
0.36 551 



0.36 516 
0.36481 

0.36447 

0.36 412 
0.36 377 



0.36 343 
0.36308 
0.36273 
0.36239 
o. 36 204 



0.36 170 
0.36 135 
0.36 lOI 
o. 36 065 
0.36 032 

o- 3 5 997 
03 5 963 
0-35928 
0.35894 

0-35859 



0.35825 
0.35791 

0.35 756 
0.35 722 

0.35688 



0-35653 
0.35619 

0.35585 
0.35551 
0-35 517 



0.35482 

0.35 448 
0.35414 
0.35 380 
0.35 346 



0.35 312 
0.35 278 
0.35244 
0.35 209 
0-35 i7l 
0.35 141 

Loir. Tsui. 



Locr. Cos. 



96 402 

96 397 
96 392 

96386 
96381 



96375 
96370 

96365 
96359 
96354 



96 349 
96343 
96338 
96332 
96327 



96 321 

96316 
96 31 1 

9630I 
96 300 



96 294 
96 289 
96 283 
96278 
96 272 



96 267 
96 261 
96 256 
96 251 
96245 



96 240 
96234 
96 229 
96 223 
96 218 



96 212 
96 205 
96 201 



96 

96 



96 
96 
96 
96 
96 



96 
96 
96 
96 
96 



96 
96 
96 
96 
96 



95 
90 



84 
79 
73 
68 
62 



57 

51 
46 
40 
34 



29 
23 
18 
12 
06 



96 lOI 
96095 
96 090 
96 084 
96078 
9-96073 
Lost. Sin. 



«0 

59 
58 

57 
56 



54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



25 
24 

23 
22 
21 



20 

19 
18 

17 
16 



15 
14 

13 
12 

II 



10 

9 

8 

7 
6 



I'. 1' 



6 

7 
8 

9 
10 

20 
30 
40 
50 



6 

7 
8 

9 
10 
20 
30 

40 
50 



3S 

3.5 

4.1 

4-7 

5-3 

5-9 

II. 8 

23-6 
29.6 



35 

3 

4 

4 

5 

5 
1 1 

17 
23 
29 



34 


34 


3-4 


3.4 


4.0 
4.6 


3 
4 


9 

5 


5.2 


5 
5 


I 
6 


II. 5 


1 1 


3 


17.2 


17 





23-0 

28.7 


22 
28 


6 
3 



6 

7 
8 

9 
10 

20 

30 

40 

50 



30 

3-0 

3.5 
4.0 

4-5 
5.0 

lo.o 

15.0 

20.0 
25.0 





29 


29 


28 


6 


2.9 


2.9 


2. 


7 


3-4 


3-4 


3- 


8 


3-9 


3-8 


3- 


9 


4.4 


4-3 


4- 


10 


4.9 


4-8 


4- 


20 


9-8 


9-6 


9- 


30 


14-7 


14.5 


14. 


40 


19-6 


19-3 


19. 


50 


24.6 


24.1 


23- 



6 


6 

0.6 


5 
0-5 


7 
8 


0.7 
0.8 


0.6 


9 
10 


0.9 

I.O 


0.8 
0.9 


20 


2.0 


1-8 


30 
40 
50 


3-0 
4.0 
5.0 


2.? 

3-6 
4.6 



5 

0.5 
0.6 

0.6 
o.? 
0.8 

1-6 
2.5 

3-3 
4.1 



r. r 



66 



371 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

24° 



29 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 

54 



55 
56 
57 
58 
59 
60 



Loff. Sin. 



9.60931 
9.60959 
9.60988 



d. 



772 
800 
828 
856 
883 



911 

938 
966 

994 
9.62 021 



9.62 049 
9.62075 
9.62 104 
9.62 13T 
9.62 158 



9.62 186 
9.62 213 
9.62 241 
9.62 268 
9.62 295 



9.62323 
9.62 350 
9.62 yj^ 
9.62 404 
9.62432 



9.62459 
9.62486 
9.62 513 
9.62 540 
9.62 56^ 

9>62 595 

Log. Cos. 



28 

28 

28 

28 



28 

28 
28 
28 

28 

28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 

27 
28 
27 
28 

2^ 
28 

2f 
28 
27 

2^ 
2^ 

27 
28 
27 
27 
2? 
27 
2f 
27 
2f 
if 

27 
27 

2^ 
27 
27 
27 
27 

2^ 

27 
If 
27 
27 
27 

2? 



Los, 'i'an. C. d. Log. Cot. 



9.64858 
9.64 892 
9.64926 
9.64 960 

9-64994 



9.65 028 
9.65 062 
9.65 096 
9.65 129 
9-65 163 



9.65 197 
9.65 231 
9.65 265 
9.65 299 
9-65 332 

9-65 366 
9.65 400 

9-65433 
9.65467 

9-65 501 

9-65 535 
9.65 568 
9.65 602 
9.65635 
9.65 669 



9.65703 

9-65 736 
9-65 770 
9.65 803 

9-65 837 

9.65 870 
9-65 904 
965 937 
9.65971 

9. 66 004 



9.66037 
9.66071 
9.66 104 
9.66 13^ 
9.66 171 



9.66 204 
9.66 23^ 
9.66 271 
9. 66 304 
96633^ 

9.66370 
9.66404 
9.66437 
9.66 470 
9- 66 503 



9.66 
9.66 
9.66 



536 
570 
603 



9.66 636 
9.66669 



9.66 702 
9.66735 
9.66768 
9.66 801 
9.66 834 

9^86f 

Log. Cot. c. d. 



34 

34 

33 

34 

34 

34 

34 

33 

34 

34 

33 

34 

34 

33 

34 

33 

33 

34 

33 

34 

33 

33 

33 

33 

34 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 



0.35 141 
0.35 107 

0-35073 
0.35 040 
0.35 006 



0.34972 

0.34938 
0.34904 

0,34 '^']0 
0.34836 



o. 34 802 
0.34769 

0.34735 
0.34701 
0.3466^ 



0.34633 
o. 34 600 

0.34566 

0.34532 

0.34499 



0.34465 

0.34431 
0.34398 
0.34364 
0.34331 



0.34297 
0.34263 
0.34230 

0.34196 
0.34 163 



0.34 129 
0.34096 
o. 34 062 
0.34029 
0.33996 



0.33 962 
0.33929 

0-3389? 
0.33862 
0.33829 



0.33795 
0.33762 

0.33729 
0.33696 
0.33 662 



Log. Cos. 



0.33629 
0.33 596 
0.33 563 
0.33529 
o. 33 496 



0.33463 
0.33430 
0.33397 
0.33364 
0.33331 



0.33 298 
0.33265 
0.33232 

0.33 198 
0.33 i6g 

0.33 J 32 
Log. Tan. 



9.96073 
9. 96 067 
9.96062 
9.96056 
9.96050 



9.96045 
9-96039 
9.96033 
9.96 028 
9.96 022 



9-96016 
9.96 01 1 
9.96005 

9-95 999 
9.95994 



9.95988 
9.95982 

9-95 977 
9.95971 
9.95965 



9-95 959 
9-95 954 
9-95 948 
9.95942 

9-95 937 



9-95931 
9-95925 
9.95919 

9.95914 
9.95908 



9-95 902 
9.95 896 
9.95891 
9.95885 

9-95 879 



9-95 873 
9.95867 
9.95 862 

9-95 856 
9.95850 



9.95 844 
9-95838 
9-95833 
9-95 827 
9-95821 



d. 



9.95815 
9.95809 
9.95 804 
9.95798 
9.95 792 



9-95786 
9.95786 

9-95 774 
9-95 768 
9-95 763 



9-95 757 
9-95 751 
9-95 745 
9-95 739 
9-95 733 
9-95 72^ 

Log. Sin. 



6 
6 

5 
6 
6 

I 
6 
6 
6 

I 
6 
6 
6 

I 
6 

6 
"dT 



GO 

59 
58 
57 
56 



55 
54 
53 
52 
51 



oO 

49 
48 

47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



25 
24 

23 
22 
21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 



10 

9 
8 

7 
6 



P. p. 





34 


33 


33 


6 


3.4 


3-3 


3- 


7 


3-9 


3 


9 


3- 


8 


4-5 


4 


4 


4- 


9 


5-1 


5 





4- 


10 


5-6 


5 


6 


5- 


20 


II-3 


II 


I 


II. 


30 


17.0 


16 


1 


16. 


40 


22.6 


22 


3 


22. 


50 


28.3 


27.9 


27. 



28 28 



3-2 

3-^ 
4.2 

4-6 

9- 
14. 
18. 

23- 



6 


2.8 


7 


3-3 


8 


3.8 


9 


4-3 


10 


4.7 


20 


9.5 


30 


14.2 


40 


19.0 


50 


23-/' 





2f 


6 


2.f 


7 


3-2 


8 


3-6 


9 


4.1 


10 


4.6 


20 


9.1 


30 


13-7 


40 


18.3 


50 


22.9 



27 

2.7 

3-1 
3.6 

4.6 

4-5 
9.0 

13-5 
18.0 

22.5 





6 


6 


0.6 


7 
8 


0.7 
0.8 


9 
10 


0.9 

I.O 


20 


2.0 


30 
40 
50 


3-0 
4.0 
5.0 



0.5 

0.6 

0.1 

0.8 
0.9 

1-8 
2.^ 
3-6 
4.6 



P. p. 



65 



372 



TABLE VII. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

25" 





I 

2 

3 

4 

5 
6 

7 
8 

9 

10 

II 

12 

13 

15 

i6 

17 

i8 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 
29 



ao 

31 

32 

33 

34 

35 
36 

37 
38 
39 



40 

41 

42 

43 

44 

45 
46 

47 
48 

49_ 

oO 

51 
52 
53 

54 

55 
56 
57 
58 
59 
GO 



liOir. sill. 



9.62 595 
9.62 622 
9.62 649 
9.62 676 
9.62 703 



(]. 



9.62 730 
9.62757 
9.62 784 
9.62 81 I 
9.62838 



9.62864 
9.62 89T 
9.62 9I8 
9.6294^ 
9.62 972 



9.62999 
9.63025 
9.63 052 
9.63079 
9.63 106 



9.63 132 
9.63159 
9.63 186 
9.63 212 
9.63239 



9.63 266 
9.63 292 
9-63319 

9-63 34 5 
9.63372 



9-63 398 
9.63425 

9.63451 
9.63478 
9.63 504 



9.63 530 
9-63557 
9-63 583 
9.63609 

9 63 636 



9.63 662 
9.63688 
9.63715 
9.63 741 
9.63767 



9-63793 
9.63819 
9.63 846 
9.63 872 
9.63898 



9.63 924 
9.63956 

9-63976 
9.64002 

9.64028 



9.64054 
9.64086 
9.64 106 
9.64 132 

9-64 158 
9.64 184 

Log. Cos. J d. 



27 
27 

27 
27 

27 

27 
27 

27 
27 
26 

27 
27 
27 

26 

27 

^6 
27 
26 

27 

26 

27 

26 
26 
26 

27 

26 
26 
26 
26 
26 
26 
26 
26 

26 

26 
26 
26 

26 

26 
26 

26 

26 

26 

26 

26 
26 

26 

26 
26 

26 

26 

26 
26 
26 
26 
26 
26 
26 
26 

25 



Log. Tan. I c d. I Lot?. Cot 



9.66867 

9. 66 900 
9-66933 
9.66 966 
9.66999 



9.67032 
9.67 065 
9-6709? 
9-67 130 
9-67 163 



9.67 196 
9.67 229 
9.67 262 

967294 
9.67 327 



9.67 360 

9-67393 
9-67425 
9-67458 
9.67 491 



9.67 523 

9-67 556 
9.67 589 
9.67 621 
9.67654 



9.67 687 
9.67719 
9.67752 
9.67784 
9.67 817 



9.67849 
9.67 882 
9.67 914 
9-67947 
9.67979 



9.68 012 
9.68 044 
9.68 077 
9.68 109 
9.68 14T 



9.68 174 
9.68 205 
9-68 238 
9.68 271 
9-68 303 



9-68 335 
9.68 368 
9. 68 400 
9.68432 
9.68 464 



9-68497 
9.68 529 
9.68 561 
9-68 593 
9.68625 



9.6865^ 
9.68 690 
9.68 722 

9-68754 
9.68786 

9.68818 



32 
33 
33 
33 
33 
33 
32 
33 
33 

33 
32 
33 
32 
33 
32 
33 
32 
33 
32 

32 

33 
32 

32 
33 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 



Log. Cot. I c. (1. 



0.33 132 
0.33 100 
0.33067 

0-33034 
0.33001 



0.32 968 

0-32935 
0.32 902 

0.32 869 
0.32836 



0.32 803 
0.32 771 
0.32738 
0.32 705 
0.32 672 



0.32 640 
0.32 607 
0.32 574 
0.32 54T 
0.32 509 



0.32476 
0.32443 
0.32 41 I 

0.32 378 
0.32 345 



0.32 313 
0.32 286 
0.32 248 
0.32 215 
0.32 183 



0.32 150 
0.32 118 
0.32 085 
0.32053 
0.32 026 



0.3 
03 
0.3 
0.3 
0.3 



0-3 
0.3 

0.3 

3 



0.3 
0.3 
0.3 
0.3 
0.3 



0.3 
0.3 
0.3 
0-3 
0.3 

0.3 



988 

955 
923 
891 

858 



826 

793 
761 

729 
6% 
66if 
632 
600 

56? 
535 



503 
471 
439 
406 

374 



342 
310 
278 
246 
214 
T82" 



Log. Tan. 



Locr. Cos. 



(1. 



9.95 727 
9.95721 
9.95716 
9.95710 
9.95704 



9.95698 
9.95692 

9.95 686 
9.95 686 
9.95674 



9.95668 
9.95 662 

9-95656 
9.95656 

9-95644 



9-95 638 
9.95632 

9.95627 

9.95 621 

9.95615 



9.95609 
9.95 603 
9-95 597 
9-95 591 
9-95 585 



9-95 579 
9-95 573 
9.95 567 
9.95 561 
9-95 555 



9-95 549 
9-95 543 
9-95 537 
9-95 530 
9-95 524 



9 95 518 
9.95512 

9-95 506 
9 95 506 
9-95 494 



9-95488 
9.95482 
9.95476 
9.95 470 
9.95464 



9-95458 
9.95452 
9-95 445 
9-95 439 
9-95 433 
9.95427 
9.95421 
9.95415 
9-95409 
9-95403 



9-95 397 
9-95 390 
9.95 384 
9-95 378 
9-95 372 
9-95 366 

Log. Sin. I" 



(iO 

59 
58 
57 
56 



30 

29 

23 

27 

26 

25 
24 

23 
22 

21 



20 

^9 
18 

17 

_i6 

IS 
14 

13 
12 

II 

9 
8 

7 
6 



p. I*. 



27 



6 

7 
8 

9 
10 

20 

30 
40 

50 



18 



22. ^ 





33 


32 


32 


6 


3-3 


3-2 


3-2 


7 


3-8 


3-8 


3-^ 


8 


4.4 


4-3 


4.2 


9 


4-9 


4.9 


4-8 


10 


5-5 


5-4 


5-3 


20 


II. 


10.8 


10.6 


30 


16.5 


16.2 


16.0 


40 


22.0 


21.6 


21.3 


50 


27.5 


27.1 


26.6 





2S 


26 


25 


6 


2.6 


2.6 


2. 


7 
8 


3 
3 


I 


3 
3 




4 


3- 
3- 


9 


4 





3 


9 


3- 


10 

20 


4 
8 


4 
8 


4 
8 


3 
6 


4- 
8. 


30 


13 


2 


13 





12. 


40 


17 


6 


17 


3 


17. 


50 


22 


I 


21 


6 


21. 



6 

7 
8 

9 
10 

20 
30 
40 
50 



8 6 5 

0.5 
0.6 
0.7 
0.8 
0.9 
1-8 

2.f 

3-6 
4-6 



0.6 


0.6 


0.7 


0.7 


0.8 


0.8 


I.O 


0.9 


1. 1 


1.0 


2.1 


2.0 


3-2 


3-0 


4-3 


4.0 


5-4 


5.0 i 



P. 1'. 



G4' 



373 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 

26° 







10 

II 

12 

14 



15 
i6 

17 

i8 

19 



20 

21 

22 

23 

24 



25 
26 

27 
28 
29 



30 

31 

32 
33 

34 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 
GO 



Lo^. Sill. d. 



64 184 
64 210 
64236 
64 262 
64 287 



64313 
64339 
64365 
64391 
644I6 



64442 
64468 

64493 
64519 

64545 



64576 

64596 
64622 

6464^ 
64673 



64698 

64724 

64749 

64775 
64 800 



64826 
64851 
64876 
64902 
64927 



64952 
64978 
65 003 
65028 
65054 



65079 
65 104 
65 129 

65 155 
65 180 



65 205 
65 230 

6525.^ 
65 286 

65305 



65331 
65356 
65381 
65 406 

65431 



65456 
65481 
65 506 
65530 
65555 



65 586 
65605 
65 630 
65655 
65 680 

9^65704 



26 
26 
26 

25 
26 
26 

25 
26 

25 
26 

25 
25 
26 

25 
25 
25 
26 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
24 

25 
25 
25 
24 
25 
25 
24 



Log. Cos. i d. 



Loff. Tan. c. d 



69615 
69647 

69678 
69 716 

69742 



68818 
68850 
68882 

68 914: 
68946 



68978 
69016 
69042 
69074 
69 106 



69138 
69 170 
69 202 
69234 
69 265 



69 297 

69329 
69 361 

69393 
69425 



69456 
69488 
69 520 
69552 
69583 



69773 
69805 

69837 
69868 
69 900 



69931 
69 963 
69994 
70026 
70058 



70089 
70 121 
70152 
70183 
70215 



70246 
70278 
70309 

70341 
70372 



70403 
70435 
70466 
70497 
70529 



70 566 

70591 
70623 

70654 

70685 

70716 
Log. Cot. 



32 
32 
32 
32 
32 
32 
32 
31 
32 

32 
32 
32 
32 
31 
32 
32 
31 
32 
32 

31 

32 

31 

32 

31 

32 

31 

31 

32 

31 

31 

32 

31 

31 

31 

31 

31 

31 

32 

31 

31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 

TTd. 



Log. Cot. 



0.31 182 
0.31 150 
0.31 11^ 
0.31085 
0.31053 



0.31 021 
0.30989 

0.3095? 
0.30 926 
0.30894 



0.30 862 
0.30830 
0.30798 
0.30 766 
0.30734 



o. 30 702 
0.30 676 

0.30639 

o. 30 607 

0.30575 



0.30543 
0.30 511 

0.30480 
o. 30 448 

0.30 4I6 



0.30384 

0.30353 
0.30321 

0.30 289 
0.30 258 

0.30226 
0.30194 
0.30163 
0.30 1 31 

0.30 100 



0.30068 
0.30037 

0.30005 

0.29973 

0.29 942 



0.29 910 
0.29 879 
o. 29 847 

0.29 8 16 
0,29 785 



0.29753 
0.29 722 
o. 29 696 
0.29 659 
0.29 628 



0.29596 

0.29 565 

0.29533 

0.29 502 
0.29471 



Log. Cos. 



0.29439 
O.29408 
0.29377 
0.29346 
0.29314 
0.29 283 
Log. Tan. 



95366 
95360 

95 353 
95 34? 
95341 



95 335 
95329 
95323 
95316 
95310 



95304 
95298 
95292 
95285 
95279 



95273 
95267 
95 266 

95254 
95248 



95242 

95235 
95229 

95223 
95 217 



95 210 
95204 



95 

95 
95 



95 
95 
95 
95 
95 



95 
95 
95 
95 
95 



98 

91 

85 



79 
73 
66 
60 

54 



47 
41 
35 
28 
22 



16 

09 
03 



95097 
95090 



95084 
95078 
95071 
95065 
95058 



95052 
95046 
95039 
95033 
95026 



95 020 
95014 
9500? 
95 001 

94 994 
9-94988 

Log. Sin. 



d. 



00 

59 
58 
57 

55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 

24 

23 

22 

21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 



10 



p. p. 





32 


6 


3-2 


7 


3.8 


8 


4-3 


9 


4.9 


10 


5.4 


20 


10.8 


30 


16.2 


40 


21.6 


50 


27.1 



32 

3-2 

3-7 

4.2 

4.8 

5-3 

10.6 

16.0 

21.3 

26. s 



6 

7 
8 

9 
10 

20 
30 
40 
50 



31 

3-1 
3-7 
4.2 

4-7 

5.2 

10.5 

I5-? 
21.0 

26.2 



31 

3-1 



3 
4 

4 

5 
10 

15 
20 

25 





26 


25 


6 


2.6 


2.5 


7 


3-0 


3-0 


8 


3-4 


3-4 


9 


3-9 


3-8 


10 


4.3 


4.2 


20 


8.6 


8.5 


30 


13.0 


i2.^ 


40 


17-3 


17.0 


50 


21.6 


21.2 





24 


§ 


6 


2.4 


0.6 


7 


2-8 


0.7 


8 


3-2 


0.8 


9 


37 


i.o 


10 


4.1 


I.I 


20 


8.T 


2.1 


30 


12.2 


3-2 


40 


16.3 


4-3 


50 


20.4 


5-4 



25 

2.5 
2.9 

3-3 

3-? 
4.1 

8.3 
12.5 

16.6 
20.8 



6 

0.6 
0.7 
0.8 
0.9 
1.0 
2.0 
3.0 
4.0 
5.0 



P. P. 



■ o 



374 



TABLE VII. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 



Lo;?. Sin. 



9.65 704 
9.65729 

965754 
9.65779 

9-65 80 3 
9.65828 
9.65853 
9.65 878 
9.65 902 
9.65927 



9.65 95T 

9-65 976 
9.66001 
9.6602^ 

9.66 050 



9.66074 
9. 66 099 
9.66 123 
9.66 1 48 
9.66 172 



9 66 197 
9.66 221 
9.66 246 
9.66 276 
9.66 294 

9.66 319 

9-66343 
9.66 367 
9.66 392 
9.66416 



9. 66 440 
9.66 465 
9.66489 
9.66513 
9.66 537 



9.66 561 
9.66 586 
9.66 610 
9.66634 
Q 66 658 



9.66682 
9-66705 
9.66 730 
9.66754 
9.66778 



9.66 802 
9.66 825 
9.66 850 
9.66874 
9.66898 



9.66 922 
9.66946 
9.66 976 
9-66994 
9 67 018 



(1. 



9.67 042 
9.67 066 
9.67089 
9.67 1 13 
9-67 137 
9.67 161 
Log. Cos. I (1. 



25 
24 

25 
24 

25 
24 

25 
24 

24 
24 
25 
24 
24 

24 
24 

24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 

24 
24 
23 
24 
24 
24 
23 
24 
23 
24 



liOff. Tan. c. d. 1 Lot?, lot. 



70716 
70748 
70779 
70810 
70 841 



70872 
70903 

70935 
70 966 
70997 



028 
059 
090 
121 
152 



183 
214 

24^ 

276 
307 



338 
369 
400 

431 
462 



493 
524 

555 

586 

617 



647 

678 
709 
740 

771 



80 f 
832 
863 
894 
925 



955 
986 
72 017 
72047 
72 078 



72 109 
72 139 
72 170 
72 201 
72231 



72 262 

72 2G2 
72323 
72354 
72384 



72415 

7244? 
72476 
72 5O6 

72 537 
9-72 567 

Lot,'. Cot. 



31 
31 
31 



31 
31 
31 
31 



o' 

^I 



31 
31 
31 
30 
31 
31 
31 
30 
31 
31 
30 
31 
30 

31 
31 
33 
31 
33 
30 
31 
30 
31 
30 
30 
30 
31 
30 
30 
30 
30 
31 
30 
30 
30 
30 
30 
30 
30 



0.29 283 
0.29 252 
0.29 221 
0.29 190 
0.29 158 



0.29 127 
o. 29 096 
0.29065 
0.29034 
o. 29 003 



0.28 972 
0.28 946 
0.28 909 
0.28 878 

0.28847 



0.28 815 

0.2878^ 
0.28754 

0.28 723 
0.28 692 



0.28 661 
0.28 636 
0.28 599 
0.28 568 
0.28537 



0.28 505 
0.28 476 
0.28 445 
0.28 414 
0.28383 



0.28 352 
0.28 321 
0.28 290 
0.28 260 
0.28 229 

0.28 198 
0.28 167 
0.28 136 
0.28 106 
0.28 075 



o. 28 044 
0.28 014 
0.27983 
0.27 952 
0.27 921 



0.27 891 
0.27 860 
0.27 830 
0.27799 
0.27 768 



0.27 738 
0.27707 
0.27 677 
0.27 646 
0.27 615 



0.27 585 
0.27 554 
0.27 524 
0.27493 
0.27 463 

0-27432 

c. (1. I Log. Tun. 



Lot;. Co.s. 



94988 
94981 

94 975 
94969 
94962 



94956 
94 949 
94 943 
94 936 
94930 



94923 
94917 
94910 
94904 
9489? 



94891 
94884 
94878 
94871 
94865 



94858 
94852 

94845 
94839 
94832 



94825 

94819 
94812 
94806 
94 799 

94 793 
94786 
94 779 
94 773 
94766 



94760 
94 753 
94 746 
94740 

94733 



94727 
94720 

94713 
94707 
94706 



94693 
94687 
94680 

94674 
94667 



94 660 
94654 
94647 
94 646 

94633 
94627 
94 620 
94613 
94 607 
94 600 

9-94 593 

Loir. Sin. 



00 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
3' 



30 

29 
28 

27 
26 

~^ 
24 
23 

21 



20 

19 

18 

17 

16 



15 
14 

13 
12 

1 1 

To 

9 

8 

7 
6 



r. i\ 





3 


I 


3 


I 


6 


3-1 


3-1 


7 


3 


7 


3 


6 


8 


4 





4 


I 


9 


4 


7 


4 


6 


10 


5 


-7 


5 


I 


20 


10 


5 


10 


3 


30 


15 


1 


15 


5 


40 


21 





20 


6 


50 


26 


2 


25 


8 



25 



6 


2. 


7 


2. 


8 


J- 


9 


3- 


10 


4- 


20 


8. 


30 


12. 


40 


16. 


50 


20. 



30 

3-3 
4.6 

4-6 

5-1 

10. r 
15.2 
20.3 
25.4 





24 


24 


2j 


6 


2.4 


2.4 


2. 


7 
8 

9 


2 



3 


8 
2 

7 


2.8 
3-2 
3-6 


2. 

3- 


10 
20 


4 
8 


I 
I 


8.0 


3- 

7- 


30 


12 


2 


12.0 


11. 


40 


16 


3 


16.0 


15- 


50 


20 


4 


20.0 


19. 



6 

7 
8 

9 
10 

20 

3^ 
40 

50 



I'. !• 



0.7 


0.6 


0.6 


0.8 
0.9 


0.7 
o-S 


0.7 
o.S 


I.O 


1.0 


0.9 


i.T 


1. 1 


1.0 


2.3 
3-5 
4.6 


2.1 
3-2 
4-3 


2.0 

3-0 
4.0 


5-8 


5-4 


5.0 



63 



375 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

28" 



15 
16 

17 
18 

19 



20 

21 

22 

23 

I ^ 

25 
26 

27 

! 28 
29 

30 

31 

32 
33 
34 

35 
36 

37 
38 
39 
40 

41 
42 
43 
44 

45 
46 

47 
48 

49 
50 

! 51 
52 
53 

54 

55 
56 
57 
58 
59 
60 



Los. Siu. 



9.67 161 
9.67 184 
9.67 208 

9.67 232 

9.67 256 



9.67279 
9.67 303 
9.67 327 
9.67356 

9-67 374 



9.67 397 
9.67421 
9.67445 
9.67468 
9.67492 



9.67515 

9-67 539 
9.67 562 
9.67 586 
9.67 609 



967633 
9.67 656 
9.67679 
9.67703 
9.67 726 



9.67 750 
9.67773 
9-67 796 
9.67 819 
9.67 843 



9.67 866 
9.67889 
9.67913 
9.67936 
9.67959 



9.67 982 
9.68005 

9.68 029 
9.68 052 
9.68075 



9.68098 
9.68 12T 
9.68 144 
9.68 167 
9.68 190 



9.68 213 
9.68236 
9 68 259 
9.68282 
9.68305 



9.68328 
9.68351 

9.68374 
9.68 397 
9.68 420 



9.68443 
9.68466 
9.68488 
9.68 51T 
9-68 534 
9-68 557 
Log. Cos. 



d. 



23 
24 
23 
24 

23 
23 
24 
23 
23 
23 
24 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 

23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
22 

23 
23 

23 

22 

23 
23 
22 

"dT 



Loff. Tail. c. d 



9.7256; 
9.72 598 
9.72628 
9.72659 
9.72 689 



9.72719 
9.72750 
9.72 780 
9.72 811 
9.72841 



9.72 871 
9.72 902 
9.72932 
9.72 962 
9.72993 



9.73023 

9-73053 
9.73084 

9-73 114 
9-73 144 



9-73 ^74 
9.73205 

9-73235 
9.73265 

9-73295 



9.73 325 
973356 
9-73386 
9.73416 

9-73 446 



9-73 476 
9-73 506 
9-73 536 
9-73567 
9-73 597 



9.73627 

9-73657 
9-73687 
9-73717 
9-73 747 



9-73777 
9.73807 

9-73837 
9.73867 

9-73897 



9.73927 
9.73957 
9-73987 
9.74017 
9.74047 



9- 74076 
9.74 106 

9-74 136 
9.74 166 
9.74 196 



9.74 226 
9-74256 
9.74 286 

9-74315 
9-74 34g 

9-74 375 
Log. Cot. 



30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
29 
30 
30 
30 
29 
30 
30 
30 
29 
30 
29 



Log. Cot. 



0.27432 
0.27 402 
0.27371 
0.27341 
0.27 311 



0.27 286 
0.27 250 
0.27 219 
0.27 189 
0.27 159 



0.27 128 

0.27098 
0.27067 

0.2703; 

0.27 007 



0.26976 
0.26946 

0.26 916 

0.26886 
0.26855 



0.26 825 
0.26 795 
0.26 765 
0.26734 
o. 26 704 



0.26674 
o. 26 644 
0.26 614 
0.26 584 

0.26553 



0.26 523 

0.26493 

0.26463 

0.26433 

0.26 403 



0.26373 
0.26343 
0.26313 

0.26 283 
0.26 253 



0,26 223 
0.26 193 
0.26 163 

0.26 133 

0.26 103 



0.26 073 
o. 26 043 
0.26 013 
0.25 983 

0.25953 



0.25923 
0.25893 

0.25 863 

0.25 833 

0.25 804 



0.25774 
0.25744 
0.25 714 

0.25 684 

0-25654 

0.25 625 



c. d. i Log. Tan. 



Los. Cos. 



9.94 593 
9.94587 
9.94580 
9.94 573 
9.94 566 



9.94560 

9 94 553 
9-94 546 
9-94 539 
9-94 533 



9.94 526 
9.94519 
9.94512 
9.94 506 
9-94 499 



9.94492 
9-94485 
9-94 478 
9-94472 
9-94465 



9-94458 
9-94451 
9-94 444 
9-94 437 
9-94 431 



9.94424 
9.94417 
9.94416 
9-94403 
9-94 396 
994390 
9-94383 
9-94376 

9-94369 
9.94 362 



9-94 355 
9-94 348 
9-94 341 

9-94 335 
9.94328 



9.94321 
994314 
9-94307 
9.94306 

9-94295 



9.94286 
9.94279 
9.94272 
9.94265 
9-94258 



9.94251 

9-94 245 
9.94238 
9.94231 
9.94224 



9.94217 
9.94 210 
9-94203 
9.94 196 
9-94 189 
9.94 182 

Log. Siu. 



45 
44 

43 
42 

41 



40 

39 
38 
37 
36 



35 
34 

33 
3^- 
31 



30 

29 

28 

27 
26 



25 
24 

23 
22 

21 



20 

19 
18 

17 
16 



15 
14 

13 
12 
II 



10 

9 
8 

7 
6 



P. P. 



30 30 29 



6 

7 
8 

9 
10 

20 

30 

40 

50 



24 

2.4 

2.8 

3-2 

3-6 

4.0 

8.0 

12.0 

16.0 

20.0 



6 


3.0 


3-0 


-7 


7 


3-5 


3-5 


3- 


8 


4.0 


4.0 


3- 


9 


4-6 


4-5 


4- 


10 


5-1 


5.0 


4- 


20 


10. 1 


lo.o 


9- 


30 


15.2 


15.0 


14. 


40 


20.3 


20.0 


19. 


50 


25.4 


25.0 


24. 





23 


23 


6 


2-3 


2.3 


7 


2.7 


2.7 


8 


3-1 


3-0 


9 


3-5 


3-4 


10 


3-9 


3-8 


20 


7-8 


7-6 


30 


II. 7 


II. 5 


40 


15-6 


15-3 


50 


19.6 


19. 1 



22 

2.2 
2.6 

3-0 

3-4 

3-? 

7.5 
II. 2 

15.0 

18.7 



6 

7 
8 

9 
10 
20 

30 
40 

50 



7 

0.7 
0.8 
0.9 
1.6 
i-i 
2.3 
3-5 
4-6 
5-8 



0.6 
0.7 

0.8 
i.o 
I.I 

2.t 

3-2 

4.3 

5-4 



P. P. 



Gl 



376 



TABLE VII. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND CO TANGENTS, 

2\y 





I 

2 

4 

5 

6 

7 
8 

9 

10 

II 

12 

13 
14 

15 
i6 

17 
i8 

19 

20 

21 

22 

23 

24 



-3 
26 

27 
28 

29 

;io 

3^ 
32 
33 

34 



35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 
00 



liOc:. Sin. 



9.68557 
9.68 580 
9.68 602 
9.68625 
9.68648 



9.68671 
9.68 693 

9-68716 
9.68739 
9.68 76T 



d. 



9.68 784 
9.68807 
9.68 829 
9.68852 
9.68 874 



9.68897 
9.68 920 
9.68 942 
9.68 965 
9.68 987 



9.69 010 
9.69 032 
9.69055 
9.69077 
9.69099 



9.69 122 
9.69 144 
9.69 167 
9.69 189 
9.69 211 



9.69 234 
9.69 256 
9.69278 
9.69 301 
9.69323 



9 69 34^ 
9.69367 
9.69390 
9.69412 
9-69434 



9-69 456 
9.69478 
9.69 506 
9.69523 
9.69 545 



9-69 567 
9-69 589 
9.69 61T 
9.69633 
9.69655 



9.69677 
9.69699 
9.69 721 

9-69743 
9.69765 

9-69787 
9.69 809 
9.69 831 
9.69853 
9-69875 

9-69897 
Log. Cos. 



23 

22 

-3 
22 

23 

23 



23 

22 

22 

22 

22 

22 

23 
22 

22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 

22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
21 
22 
22 

"dT 



Los. Tan. led. 



9-74 375 
9.74405 

9-74 435 
9.74464 

9-74 49-1 



9.74524 
9.74554 

9-74583 
9.74613 

9- 74 643 



9.74672 
9.74702 

974732 
9.74761 
9.74791 



9.74821 
9-74850 
9. 74 880 
9.74909 
9-74 939 



9-74969 
9-74 998 
9.75 028 
9.75057 
9.75087 



9-75 116 
9.75 146 

9-75 17^ 
9.75205 

9-75 234 



9.75 264 
975293 
975323 
9-75352 
9.75382 



9.75 411 
9.75441 
9.75470 

9-75 499 
9.75 529 



9-75 558 
9.75 588 
9.75617 

9.75 646 
9.75676 



9.75705 

9-75 734 

9-75764 

9-75 793 
9.75 822 



9.75851 
9.75881 
9.75910 

9-75 939 
9.75968 



9-75998 
9.76027 
9.76056 
9.76085 
976 115 
9.76 144 



30 
30 
29 
30 

29 
30 
29 
29 
30 

29 
30 
29 
29 

30 

29 
29 
29 
29 
30 
29 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 
29 

29 

29 
29 

29 
29 
29 
29 
29 
29 
29 

29 

29 
29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 



Log. Cot. I c. d. 



Lo«?. Cot. 


l-(u:. Cos. 


d. 

7 
7 
7 
7 
7 
7 
7 
7 
? 
7 
7 
7 
7 
7 
7 
7 
1 
7 
7 
7 
7 
7 
7 
7 
7 
1 
7 
7 
7 
1 
7 
7 

1 

7 




0.25625 


9.94 182 


()0 


0.25 595 


9.94175 


59 


0.25 565 


9.94 168 


58 


0-25 53^ 


9.94 161 


57 


0.25505 


9.94154 


56 


0.25476 


9.94 147 


55 


0.25 446 


9.94 140 


54 


0.25 416 


9-94133 


53 


0.25387 


9.94126 


52 


0.25357 
0.25 327 


9-94 118 


51 
50 


9.94111 


0.25 297 


9.94 104 


49 


0.25 268 


9-9409? 


48 


0.25 238 


9-94090 


47 


0.25 208 


9.94083 
9- 94076 


46 
45 


0.25 179 


0.25 149 


9.94069 


44 


0.25 120 


9.94062 


43 


0.25 090 


9.94055 


42 


0. 2 5 060 


9.94048 


41 


0.25 031 


9-94041 


40 


0.25 001 


9-94034 


39 


0.24972 


9-94026 


3a 


0.24 942 


9.94019 


37 


0.24913 


9.94012 


36 

35 


0.24883 


9.94005 


0.24854 


9.93998 


34 


0.24 824 


9.93991 


33 


0.24795 


9-93984 


32 


0.24 765 


9-93 977 


3» 


0.24736 


9.93969 


ao 


0.24 706 


9.93 962 


29 


0.24677 


9-93 955 


28 


0. 24 64^ 


9-93948 


27 


0.24 618 


9-93941 


/ 

7 


26 

25 


0.24588 


9-93 934 


0.24559 


9.93926 


7 
7 
1 
7 
7 
1 
7 
7 
7 
1 
7 
1 
7 

7 
1 
7 
1 
1 
7 
1 
7 
1 


24 


0.24529 


9-93919 


23 


0.24 500 


9.93912 


22 


0.24471 


9-93905 


21 


0.24441 


9.93898 


20 


0.24 412 


9.93891 


J9 


0.24383 


9-93883 


18 


0.24353 


9-93876 


17 


0.24324 


9.93 869 
9.93 862 


i« 


0.24 295 


15 


0.24 265 


9-93854 


14 


0.24 236 


9-9384? 


13 


0.24 207 


9.93840 


12 


0.2417^ 


9-93833 


11 
10 

9 


0.24 148 
0.24 119 


9.93 826 
9-93 818 


0. 24 090 


9-93811 


8 


0. 24 066 


9.93 804 


7 


0.24031 
0.24002 


9-93 796 


6 


993789 


5 


0.23973 


9.93782 


4 


0.23943 


9-93 775 


3 


0.23914 


9.93767 


2 


0.23885 


9-93766 


7 
1 
d. 


I 


0.23 856 


9-93 753 





Loff. Tun. 


!.(!g. sill. 


/ 



r. I* 



7 
8 

9 
10 

20 
30 
40 
50 



30 

3.0 

3.5 
4.0 

4.5 
5.0 

lo.o 
15.0 
20.0 
25.0 



29 

2.9 
3-4 
3-9 
4 4 
4.9 

9.8 
14.? 
19.6 
24.6 





23 


6 


2-3 


7 


2 


7 


8 


3 





9 


3 


4 


10 


3 


8 


20 


7 


6 


30 


1 1 


5 


40 


15 


3 


50 


19 


I 





22 


22 


6 


2.2 


2.2 


7 


2.6 


2.5 


8 


3.0 


2.9 


9 


3-4 


3-3 


10 


3-7 


3-6 


20 


7.5 


7.3 


30 


II. 2 


II. 


40 


15.0 


14-6 


50 


18.7 


18.3 



29 

2.9 



3 
3 

4 

4 

9 

14 

19 

24 



21 

2.T 

2.5 
2-8 

3.2 

3.6 

7-1 
10.? 

14-3 
17.9 



6 

7 
8 

9 
10 

20 

30 
40 

50 



1 

0.1 
0.9 
i.o 
1. 1 

1.2 
2.5 
3-? 
5.0 
6.2 



7 
0.7 
0.8 
0.9 
1.6 
i.i 
2-3 
3-5 
4-6 
5.8 



p. P. 



GO 



377 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 



10 

1 1 

12 

13 

14 



15 

16 

17 



19 



-:> 
26 
27 
28 

29 



ao 

31 
32 

33 
34 



35 
36 

37 
38 
39 



40 

42 

43 
44 



55 
56 
57 
58 
59 
<>0 



30 



20 


9 


21 


9 


22 


9 


23 


9 


24 


9 



Lost. sin. 



69897 
69919 
69 940 
69 962 
69 984 



70006 
70028 
70050 
70071 
70093 



70 115 

70137 

70158 
70 180 
70 202 



70 223 
70245 
70 267 
70288 
70310 



70331 
70353 
70375 
70396 
70418 



70439 
70461 
70482 
70504 
70525 



70547 

70568 
70590 
70 61 1 
70 632 



70654 
70675 
70696 
70718 

70739 



70 760 
70782 
70803 
70 824 
70846 



70867 
70888 
70 909 

70930 
70 952 



70973 
70994 
7101I 

71036 
71 05^ 



71 078 
71099 
71 121 
71 142 
71 163 
71 184 
hog. Cos. 



d. 



22 
21 
22 
22 
21 
22 
22 
21 
22 
21 
22 
21 
21 
22 
21 
21 
22 
21 
21 
21 
22 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 

"(IT 



Loir. Tan. 



76 144 
76 173 

y6 202 
76 231 
76 266 



76 289 
76319 
76348 
76377 
76406 



76435 
76464 

76493 
76 522 

76551 
76580 
76 609 
76638 
76667 
76696 



76725 

76754 
76783 
76812 
76 841 



76 870 
76899 
76 928 

76957 
76986 



77015 

77043 
77 072 

77 lOI 

77 130 



77 159 
77 188 
77217 

77245 
77274 



77303 
77332 
77361 
77389 
77418 



77 447 
77476 
77 504 
77 533 
77 562 



77 591 
77619 
77648 
77677 
77705 



77 734 
77763 
77791 
77 820 
77849 
77 87^ 
Log. Cot. 



29 
29 
29 
29 
29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 
29 

29 
29 

29 
29 

29 
29 
29 
29 
28 

29 

29 
29 
29 

28 

29 
29 
29 

28 

29 
29 

28 

29 
29 

28 

29 

28 

29 

28 

29 

28 

29 

28 

29 

28 
28 

29- 

28 
28 

29 

28 
28 

29 

28 



liOg. Cot. 



0.23 856 
0.23 827 

0.23797 
0.23768 
0.23739 



0.23 710 
0.23 681 
0.23 652 
0.23 623 
0.23 594 



0.23 565 
0.23 535 
0.23 506 
0.23477 
0.23 448 



0.23419 
0.23 396 
0.23 361 
0.23332 
0.23303 



0.23 274 
0.23245 
0.23 216 
0.23 18^ 
0.23 158 



0.23 129 
0.23 lOI 
0.23 072 
0.23043 
0.23 014 



0.22 985 
0.22 956 
0.22 92^ 
0.22 898 
0.22 869 



0.22 841 
0.22 812 
0.22 783 
0.22 754 
0.22 725 



0.22 696 
0.22 668 
0.22 639 
0.22 616 
0.22 581 



0.22553 
0.22 524 
0.22 495 
0.22 466 
0.22 438 



0.22 409 
0.22 386 
0.22 352 
0.22 323 
0.22 294 



0.22 266 
0.22 237 
0.22 208 
0.22 180 
0.22 151 

0.22 122 
Log. Tail. 



Los:. Cos. 



9-93 753 
93746 
93 738 
93731 
93724 



93716 
93709 
93 702 

93694 
93687 



93680 
93 672 
93665 
95658 
93656 



93643 
93635 
93628 
93621 

93613 



93606 
93 599 
93591 
93584 
93 576 



93569 
93562 

93 554 
93 547 
93 539 



93532 

93524 

93517 

93509 
93502 



93 495 
93487 
93480 

93472 
93465 



93 457 
93450 

93442 

93 435 
93427 



93420 
93412 
93405 
93 39? 
93390 



93382 
93 374 
93367 
93 359 
93352 



93 344 
93 337 
93329 
93321 
93314 
9-93 306 

Log. Sill. 



d. 



00 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 

28 

27 
26 



25 
24 
23 
22 
21 



20 

19 
18 

17 
16 



15 
14 
13 
12 
II 



10 

9 
8 

7 
6 



p. p. 





22 


21 


6 


2.2 


2.T 


7 


2.5 


2.5 


8 


2.9 


2.8 


9 


3-3 


3-2 


10 


3-6 


3-6 


20 


7-3 


7.1 


30 


II. 


10.7 


40 


14-6 


14.3 


50 


18.3 


17.9 



21 

2.1 

2.4 
2.8 

3-1 

3-5 

7.0 

10.5 
14.0 

i7'S 





8 


1 


6 


0.8 


O.J 


7 
8 


0.9 
1.6 


0.9 

I.O 


9 
10 


1.2 
I-.3 


I.I 

1.2 


20 


2-6 


2.5 


30 


4.0 


Z-1 


40 
50 


5.3 
6.6 


5.0 
6.2 



7 

0.7 
0.8 
0.9 
1.6 
I.I 
2.3 
3-5 
4--6 
5.8 





29 


29 


28 


6 


2.9 


2.9 


2.8 


7 


3-4 


3.4 


3-3 


8 


3-9 


3-8 


3-8 


9 


4.4 


4.3 


4-3 


10 


4.9 


4-8 


A-1 


20 


9-8 


9-6 


9-5 


30 


14-7 


14.5 


14.2 


40 


19-6 


19-3 


19.0 


50 


24.6 


24.1 


23.^ 



p. p. 



59 



378 



TABLE VII. — LOGARITHMIC SINES. COSINES, TANGENTS, AND COTANGENTS. 

«> i o 
O 1 



5 
6 

7 
8 

9 

10 

II 

12 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 
27 
28 
29 

30 

31 

32 

33 
34 



36 
37 
38 

39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 

oO 

51 
52 
53 
54 



55 
56 
57 
58 
-59_ 
00 



liOsr. Sill. 



d. 



9-7 
9-7 

9-7 
9-7 
9.7 



9.7 
9-7 
9.7 
9-7 
9-7 



9.7 
9-7 
9-7 
9-7 
9-7 



9-7 
9-7 
9-7 
9-7 
9-7 



9-7 
9-7 
9.7 
9-7 
9-7 



9-7 
9-7 
9.7 
9-7 
9-7 



9-7 
9-7 
9-7 
9-7 

9-7 



9-7 
9-7 
9-7 
9-7 
9-7 



184 
205 
226 
247 
268 



289 
310 

331 
351 

372 



393 
414 

435 

456 

477 



498 

518 
539 
560 
581 



601 
622 
643 
664 
684 



705 
726 

746 
767 
788 



808 
829 

849 
870 

891 



911 
932 
952 
973 
993 



9.72 014 
9.72034 
9.72055 
9.72075 
9.72 096 



9.72 116 

9-72 136 
9.72 157 
9.72 177 
9.72 198 



9.72 218 
9.72 238 
9.72259 
9.72279 
9.72299 



9.72319 
9.72340 
9.72360 
9.72380 
9.72 400 



9.72 421 



21 

21 

21 

21 

21 

21 

21 

20 

21 

21 

21 

20 

21 

21 

21 

20 

21 

20 

21 

26 

21 

20. 

21 

26 

21 

20 

26 

21 

20 

26 

20 

26 

21 

20 

20 

20 

26 

26 

20 

20 

20 

26 

20 

20 

20 

20 

26 

26 

26 

20 

26 

20 

20 

26 

20 

26 

20 

20 

20 

20 



I,otr. Tail. c. d. 



-otr. Cot. 



9.77^77 
9.77906 

9-77 934 

9-77963 
9.77992 



9.78 020 
9.78049 
9.78077 
9.78 106 
9-78 134 



9.78 163 
9.78 19T 
9.78 220 
9.78 248 
9-78277 



Log. Cos. i iU 



9.78305 

9-78334 
9.78362 
9.78391 
9.78419 



9.78448 

9-78476 
9.78505 

9-78533 
9.78561 



9.78590 

9-78618 
9.78647 
9-78675 
9.78703 



9.78732 
9.78 760 
9.78788 
9.78817 
9.78845 



9.78873 
9.78 902 
9.78930 

9-78958 
9.78987 



9.79015 

9-79043 
9.79071 

9.79 100 

9.79 128 



9-79156 

9-79 184 
9.79213 
9.79241 
9.79269 



9.79297 

9-79325 
9-79 354 
9.79382 
9.79410 



9-79 438 
9.79465 
9.79494 
9-79522 
9-79551 



9.79 579 



28 
28 
28 
29 

28 
28 
28 
28 
28 

28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 

28 
28 
28 
28 

28 

28 



28 

28 

28 
28 

28 

28 

28 

28 
28 

28 

28 

28 

28 

28 

28 

28 

28 

28 
28 

28 

28 

28 

28 
28 

28 

28 
28 
28 

28 

28 



O. 22 I 22 

0.22 094 
0.22 065 
0.22 037 
0.22 008 



0.2 
0.2 
0.2 
0.2 
0.2 



0.2 
0.2 
0.2 
0.2 
0.2 



0.2 
0.2 
0.2 
0.2 
0.2 



0.2 
0.2 
0,2 
0.2 
02 



0.2 
0.2 
0.2 
0.2 
0.2 



0.2 
0.2 
0.2 
0.2 
0.2 



0.2 
0.2 
0.2 
0.2 
0.2 



979 

951 

922 

894 

865 



837 

808 

780 

751 
723 



694 

666 

637 

609 

586 



552 
523 

495 

467 

438 



410 
381 
353 
325 
296 



268 

239 
211 

183 
154 



126 

098 
070 
041 
013 



0.20 985 
0.20 956 
0.20 928 
o. 20 900 
0.20 872 



0.20 843 
o. 20 8 1 5 
0.20 787 
0.20 759 
0.20 731 



0.20 702 
0.20 674 
o. 20 646 
0.20618 
0.20 590 



0.20 561 
0.20533 
0.20 505 
0.20477 
o. 20 449 



0.20 421 



l.oir. Cos 



liO^. C<»t. ' c. «1. I, otr. Tiin. 



93 306 
93 299 

93291 

93 284 

93276 



93268 

93 261 
93253 
93245 
93238 



93230 
93 223 
93215 
93207 
93 200 



93 192 
93 184 
93 ^77 
93 169 
93 161 



93 153 
93 146 
93 138 
93 ^'30 
93 123 



93 115 
93 107 
93 100 
93 092 
93084 



93076 
93069 
93 061 
93053 
93045 



93038 
93030 
93 022 
93014 
93 006 



92 999 
92991 

92983 
92975 

9296? 



92 960 
92952 
92944 
92936 
92928 



92 920 
92913 
92905 

92 897 
92889 



92 881 
92873 
92865 
92858 
92 850 

92 842 



9 
9 
9 
9 

9_ 

LoJT. sill. 



<l. 



GO 

59 
58 
57 

Ji 
55 
54 
53 
52 

."iO 

49 
48 

47 
46 



45 
44 
43 

42 
41 



30 

29 
28 
27 
26 



24 

23 
22 
21 



20 

'9 
18 

17 
16 



15 
14 

13 
12 
1 1 



10 

9 
8 

7 
6 



V. V 





w29 


28 


6 


2.9 


2-8 


7 


3-4 


3-3 


8 


3-8 


3-8 


9 


4-3 


4-3 


10 


4-8 


4-7 


20 


9-6 


9-5 


30 


14.5 


14.2 


40 


19.3 


19.0 


50 


24.1 


23.7 



28 

2.8 

3-2 

3-^ 
4.2 

4^ 

9-3 
14.0 

18.6 

23-3 





21 


26 


6 


2.1 


2.C 


7 


2.4 


2.4 


8 


2.8 


2.? 


9 


3-1 


3-1 


10 


3-5 


3-4 


20 


7.0 


6-8 


30 


10.5 


10.2 


40 


14.0 


13-6 


50 


17.5 


17.1 



20 

2.0 

2-3 
2.6 

3-0 

3-3 
6.6 

lO.O 

13-3 
16.6 



6 

7 
8 

9 
10 

20 

30 
40 

50 



8 

0.8 
0.9 
1.6 
1.2 
1-3 
2-6 
4-0 

5-3 
6.6 



o.f 
0.9 
i.o 
1. 1 
1.2 

2.5 

3-7 
5.0 

6.2 



I'. I' 



58' 



379 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 

32° 



10 

II 

12 

13 
14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 



30 

31 
32 
33 
34 



35 
36 
37 
38 

39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



Lo^. ISiii. 



72 421 
72441 
72 46T 
72 481 
72 501 



72 522 

72542 
72 562 
72 582 
72 602 



72 622 
72 642 
72 662 
72682 
72 702 



72 723 

72743 
72 763 
72783 
72 802 



72 822 
72 842 
72862 
72882 
72 902 



72 922 

72942 
72 962 

72 982 

73 002 



73021 

73041 
73061 
73081 

73 loi 



73 120 
73140 
73 160 
73 180 
73199 



73219 
73239 
73258 
73278 
73298 



73317 
73 33^ 
73 357 
9 73 376 
9 73 396 



73415 
73 435 
73 455 
73 474 
73 494 



73513 
73 533 
73552 
73 572 
73 59^ 
9-73611 



d. 



20 
20 
20 
20 
26 
20 
20 
26 
20 

20 
20 
20 
20 
20 
20 
20 
20 
20 

19 
20 
20 
20 
20 
20 
20 

19 
20 

20 

20 

19 

20 
20 

19 
20 

19 
20 

19 
20 

19 

20 

19 
19 

20 

19 

19 

20 

19 
19 
19 

19 

20 

19 
19 
19 
19 
19 
19 
19 
19 
19 



Lo^. Tan. c. d 



9-79 579 
9.79607 

9-79635 
9.79663 
9.7969! 



9.79719 
9-79 747 
9-79 775 
9.79803 

9-79831 



9.79859 
9.79887 
9.79915 

9.79943 
9.79971 



9.79999 
9.80027 
9.80055 
9.80083 
9. 80 1 1 1 



9.80 139 
9.8016^ 
9.80 195 
9 80 223 
9.80251 



9.80279 
9. 80 307 

9.80335 
9-80363 
9.80391 



9-80418 
9.80446 

9.80474 
9.80 502 
9-80 530 



9.80558 
9.80 586 
9.80 613 
9.80 641 
9.80 669 



9.80697 
9.80725 
9.80 752 
9.80786 
9. 80 803 



9.80836 
9.80864 
9.80 891 
9.80 919 
9.80947 



9.80975 
9.8 - 
9.8 
9.8 
9.8 



9-' 
9. J 

9-' 
9.i 

9.i 



002 
030 
058 
085 



113 
141 

168 

196 
224 

Jsl 

JiOg. Cot. 



28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

2? 
28 
28 
28 
28 

2? 
28 
28 
28 
27 
28 
28 
2f 
28 
28 

2f 
28 

2? 
28 
28 

2f 
28 

2f 
28 

27 
28 
27 
2? 

28- 

2f 
28 

2? 
29 
28 
2f 

2f 

cTJr 



Log. Cot. 



o. 20 42 1 

0.20393 

0.20 365 

0.20337 
0.20308 



0.20 286 
0.20 252 
0.20 224 
0.20 195 
0.20 168 



0.20 140 

O. 20 112 

o. 20 084 

0.20 055 
0.20 028 



o. 20 000 

0,19 972 
0.19944 
O.I99I6 
0.19 



o. 



9866 
9832 

9 804 
9 776 
9 748 



9721 

9693 
9665 

9637 
9 609 



9 581 

9 553 

9525 

9 49^ 
9470 



9442 
9414 

9386 
9 358 
9330 



9303 
9275 
924^ 
9219 
9 191 



9164 
9136 
9 log 
9 086 

9053 



9025 

899? 
8970 
8 942 
8914 



8 886 
8859 
8 831 
8803 
8 776 

8748 

Lot?. Tail. 



Log. Cos. 



9.92 842 
9.92834 
9.92826 
9.92 818 
9.92 816 



9.92 802 
9.92794 
9.92786 

9.92 778 
9.92771 



9.92763 

9.92755 

9-92747 

9-92739 
9.92731 



9.92723 
992715 
9.92707 
9.92699 
9.92 691 



9.92 683 
9.92675 
9.92 667 
9.92659 
9.92 651 



9 92 643 
9.92635 
9.92 627 
9.92 619 
9.92 611 

9.92 603 
9.92595 
9-92 587 
9-92579 
9-92 570 



9.92 562 
9.92554 
9-92 546 
9-92 538 
9-92 530 



9.92 522 
9.92514 
9.92 506 
9.92498 
9.92489 



9.92 481 

9-92473 
9.92465 
9.92457 

9-92449 



9.92441 

9-92433 
9.92424 

9.92 416 

9-92 408 



9.92 400 
9.92392 
9-92383 
9-92375 
9-92 367 

9-92359 

Log. Sin. 



d. 



00 

59 
58 

57 
56 



55 
54 
53 
52 
51 



50 

49 

48 

47 
46 



45 
44 

43 

42 

41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 

23 
22 

21 



20 

19 
18 

17 
16 



15 

14 

13 
12 

II 



10 

9 
8 

7 
6 



p. p. 





28 


28 


6 


2.8 


2.8 


7 


3-3 


3-2 


8 


3-8 


3-f 


9 


4-3 


4.2 


10 


4-7 


4-6 


20 


9-5 


9-3 


30 


14.2 


14.0 


40 


19.0 


18.6 


50 


23-7 


23-3 





20 


20 


6 


2.6 


2.0 


7 
8 


2.4 

2.f 


2.3 
2-6 


9 


3.1 


3-0 


10 

20 


3-4 
6.8 


3-3 
6.6 


30 


10.2 


lO.O 


40 
50 


13-6 
17. 1 


13-3 
16.5 





8 


8 


6 


0.8 


0.8 


7 


I.O 


0.9 


8 


I.I 


1.0 


9 


1-3 


1.2 


10 


1-4 


1-3 


20 


2-8 


2-6 


30 


4.i2 


4.0 


40 


5-6 


5-3 


50 


7.1 


6.6 



2f 

2.1 
3-2 

3-6 

4.1 

4-6 
9.1 

13-^ 
18.3 

22.9 



19 

1.9 

2-3 
2.6 

2.9 
3-2 

6.5 

9-1 
13.0 

16.2 



7 

0.1 
0.9 
1.0 
I.I 
1.2 

2-5 

3-^ 
5.0 

6.2 



P. p. 



^T 



380 



TABLE VII.— LOGARITHMIC SINES, COSINES, 



•ANGENTS, AND COTANGENTS. 



10 

1 1 

] 2 

13 

14 

15 
16 

17 

iS 

22- 
20 

21 



24 



25 
26 
27 
28 
29 

31 
32 
33 

_34_ 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 
60 

51 

52 
53 
54 

55 
56 
57 
58 

(10 



Loff. Sin. 



9.73 611 
9.73 630 
9.73650 



73 669 
73688 



73708 
73727 
73 746 
73766 

73785 



73805 
73824 
73843 
73 862 
73882 



73901 
73 926 
73940 
73 959 
73 978 



73 99? 
740^6 
74036 

74055 
74074 



74093 
74 112 

74 131 
74151 
74 170 



74189 
74 208 
74227 
74246 
74265 



74284 

74303 
74322 

74341 
74360 



74 379 
74398 
74417 
74436 
74455 
74 474 
74 493 
74 51 1 
74530 
74 549 



74568 
74587 
74606 
74625 
74643 



74 662 
74681 
74700 

74718 
74 737 
74756 

Log. Cos. 



liOt;. Tun. 



9.8 
9.8 
9.8 
9.8 

9.8 
9.8 
9.8 
9.8 
9^ 
9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 



251 

279 
307 

334 
362 



390 
417 

445 

473 
500 



528 

555 

583 
610 

638 



666 

693 
.721 

748 
776 



803 

831 
858 
886 

913 



941 

968 
996 



9.82 023 
9.82 051 



9.82078 
9.82 105 
9.82 133 
9.82 166 
9.82 188 



9.82 215 
9.82243 
9.82 276 
9.82 29^ 
9.82 325 



9.82352 
9.82 380 
9.82 407 
9.82434 
9.82 462 



9.82 489 

9-82516 
9.82 544 

9 82 571 
9.82 598 



c. (I. Loir. Cot. 



9.82 626 
9.82 653 
9.82686 
9.82708 
9.82735 



9.82 762 
9 82 789 
9.82817 
9.82 844 
9.8 2871 

9 82 898 
liOj;. Cot. 



2f 
2? 
28 

2? 
2? 
2? 
28 
27 

2f 
27 
2f 
2f 
2? 
28 

2? 
2f 
2l 
2l 
2l 
2l 
2l 
21 
2f 

2j 
2j 

2j 

2j 
2l 

2l 
27 
2l 
2l 
2l 
2j 
2j 
2l 
27 
2l 

2l 

2j 
27 

2l 
27 
27 
27 
2l 
27 
27 
2j 

27 
2l 
2l 
27 
2l 
27 
2l 
27 
2l 

27 



0.18748 
0.18 720 
0.18 693 

o.i8 66| 
0.1863^ 



Loir. Cos. 



o. 1 8 6 1 o 
0.18 582 

0.18555 

0.18 527 
o. 1 8 499 



0.18 472 
o. 1 8 444 
0.18417 
0.18389 
o. 18 362 



0.18334 
O.I8306 

0.18 279 
0.18 251 
o. 18 224 



0.18 196 
0.18 169 
0.18 I4T 
0,18 114 
o. 1 8 o8a 



0.18 059 
0.18 031 
o. 1 8 004 
0.17 976 
0.17949 



0.17 921 
0.17 894 
0.17 867 
0.17 839 
o. 1 7 8 1 2 



0.17784 
0.17757 
0.17 729 
0.17 702 
0.17675 



0.17 647 
0.17 620 
0.17 593 
0.17565 
0.17 538 



17 510 

17483 
17456 

17428 
1 7 401 



0.17 374 
0.17347 
0.17 319 
o. 17 292 
0.17 265 



0.1723? 
o. 1 7 216 

0.17 183 

0.17 156 
0.17 128 
o. 17 loT 

c. (1. I Loir. Tan. 



992359 
9.92351 
9.92342 
9-92334 
9.92 326 



9.92318 
9.92 310 
9.92 301 
9-92293 
9.92 285 



9.92 277 
9.92 268 
9.92 266 
9.92 252 
9.92 244 



9.92235 
9.92 227 
9.92 219 
9.92 216 
9.92 202 



9.92 
9.92 
9.92 
9.92 
9-92 



9.92 
9.92 
9.92 
9.92 
9.92 



94 
85 
7f 

69 
66 



52 
44 
35 
2? 
19 



10 
02 



9.92 
9.92 
9.92094 
9.92 085 
9.92077 



9.92 069 
9.92 066 
9.92 052 
9.92043 
9-92035 



9.92 027 
9.92 018 
9.92 010 
9.92 001 
9-91 993 



9-9 
9.9 

99 
9-9 
9.9 



9.9 

9-9 
9.9 

9-9 
9.9 



9-9 
9.9 

9-9 

9 9 
9.9 



984 
976 
967 
959 
951 



942 
934 

925 
917 

908 



900 
891 

883 

874 
866 



9 91 857 
Loir. sin. 



(>0 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 

48 

47 
46 



45 
44 

43 
42 

41 



40 

39 
38 
37 
36 



-:> 

24 

23 
22 
21 



10 

19 
18 

17 
16 



15 
14 

13 
12 

1 1 

9 

8 

7 
6 



r. 





28 


21 


6 


2.8 


2.1 


7 


3-2 


3.2 


8 


3-? 


3-6 


9 


4.2 


4.1 


10 


4-6 


4.6 


20 


9-3 


9.1 


30 


14.0 


13-7 


40 


18.6 


18.3 


50 


23-3 


22.9 



27 

2.7 

3-1 
3-6 
4.6 

4-5 
9.0 

13-5 
18.0 

22.5 





19 


19 


6 


1.9 


1.9 


7 


2-3 


2.2 


8 


2.6 


2.5 


9 


2.9 


2-8 


10 


3-2 


3-1 


20 


6.5 


6.3 


30 


9-? 


9.5 


40 


13.0 


12.6 


50 


li5.2 


15.8 



18 

1-8 
2.1 
2.4 
2.8 

3-1 

6.T 

9.2 

12.3 

15.4 



6 

7 
8 

9 
10 

20 
30 
40 
SO 



8 

0.8 
i.o 
I.I 

1.3 
1-4 
2-8 
4-2 
5-6 
7-1 



8 

0.8 
0.9 
1.6 
1.2 
1-3 
2.6 
4.0 

5-3 
6.6 



r. I'. 



5(j 



3*1 



iAiiLl. V 11. -LOGARITHMIC SiNES, COSINES, TANGENTS, AND COTANGENTS 



34 



24 



Log. Sill. 

9-74756 
9-74 775 
9 74 793 
9.74812 

9.74831 



9.74849 
9.74868 
9-74887 

9.7490? 
9.74924 



9-74 943 
9.74961 
9.74980 

9-74 998 
9.75017 



9-75036 

9-75054 
9.75073 
9.75091 
9-75 no 



9-75 128 
9-75 147 
9.75 165 
9.75 184 
9.75 202 



9.75 221 

9-75239 
9-75 257 
9.75276 

9-75 294 



9-75313 
9-75331 
9-75 349 
975368 
9-75386 



9-75404 
9-75423 
9-75441 
9-75 459 
9-75478 



9-75496 
9-75 54 
9-75 532 
9-75551 
9.75569 



9-75587 
9.75605 
9.75623 
9-75642 
9.75 660 



9-75678 
9-75695 
9-75714 
9-75 732 
9-75750 



9-75769 
975787 
9.75805 
9-75823 
9-75841 



975859 

liOf?. Cos. 



(1. 



19 

18 
19 
18 

18 
19 

18 

19 
18 
18 
18 
18 
18 
19 
18 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 

18 
18 
18 

18 
18 

18 
18 
18 

18 

18 
18 

18 
18 
18 

18 
18 

IS 



Log. Tan, c. d. I Log. Cot. 



9.82898 
9.82 926 
9-82953 
9.82 980 
9.83007 



9-83035 
9.83 062 
9.83 089 
9-83 116 
9-83 143 



9.83 171 
9.83198 
9.83 225 
9-83252 
9-83279 



9-83307 
9-83334 
9-83361 
9.83388 
9.83415 



9.83442 
9.83469 

9-83496 
9-83524 
9-83551 



9.83578 
9.83 605 
9-83632 
9-83659 
9-83686 



9-83713 
9.83740 

9.83767 

9-83794 
9.83821 



9-83848 
9.83875 
9.83 902 

9-83929 
9.83957 



9.83984 
9.84 01 1 
9-84038 
9.84065 
9.84091 



9.84 118 
9-84 Hi 
9.84 172 
9.84199 
9-84225 



9.84253 
9.84 286 
9-84 307 
9-84334 
9-84361 
9-84388 
9.84415 

9-84442 
9.84469 
9.84496 



27 
27 
27 
27 
27 

27 
27 

27 
27 

2f 
27 
27 
27 
27 

27 
27 
27 
27 
2f 
27 
27 
27 
27 
27 

27 
27 
27 
2? 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 

27 
26 
27 
27 
27 
27 
27 

27 
27 
27 
27 

26 

27 
27 
27 
27 
27 

26 
Log. Cot. I 0. d. 



9.84 522 



0.17 lOI 

0.17074 

0.17 047 
0.17 019 
o. 16 992 



0.16 965 
0.16938 
0.16 916 

0.16883 

0.16 855 



0.16 829 
0.16 802 

0.16774 

0.16 747 
o. 16 720 



o. 16 693 
0.16 666 
o. 16 639 
0.16 612 
0.16 584 



0.16557 
0.16 536 
0.16 503 
0.16 476 
o. 1 6 449 



0.16422 
o. 16 395 
0.16368 
0.16 346 
o. 16 313 



0.16 285 
o, 16 259 
o. 16 232 
0.16 205 
0-16 178 
0,16 151 
0.16 124 
o. 16 097 
o. 16076 
o. 16 043 



O.I60I6 
o. 1 5 989 
o. 1 5 962 

0-15935 
o. 1 5 908 



0.15 881 
0.15854 
0.15 827 
o. 1 5 800 
0.15773 



0.1 5 746 
0.15719 
o. 1 5 692 
0.15 665 

0-1563 9 
0.15 612 
0.15 585 
0.15 558 

0.15 531 
0.15504 



Log. Cos. 



9.91 857 
9.91 849 
9.91 846 
9.91 832 
9.91 823 



9.91 814 
9.91 806 
9.91 79^ 
9.91 789 
9.91 786 



9.91 772 
9.91 763 

9-91 755 
9.91 746 

9-91 737 



9.91 729 
9.91 720 
9.91 712 
9.91 703 
9.91 694 



9.91 686 
9.91 677 
9.91 668 
9.91 660 
9.91 651 



9.91 642 
9.91 634 
9.91 625 
9.91 616 
9.91 608 



9.91 599 
9.91 590 
9.91 582 

9-91 573 
9.91 564 



9.91 556 

9-91 547 

9 91 538 
9.91 529 

9.91 521 



9.91 512 
9.91 503 
9.91 495 
9.91 486 
9.91 477 



9.91468 
9.91 460 

9-91451 
9.91 442 

9.91433 



9.91424 
9.91 416 
9.91407 
9.91 398 
991 389 



9.91 386 
9.91 372 
9-91 363 
9-91 354 
9-9t 345 
0.15 47f 9-91336 

Log. Tan. | Log. Sin. 



P. P. 



6 

7 
8 

9 
10 
20 
30 
40 
50 



2f 

2.7 

3-2 

3-6 
4.1 

4.6 
9.1 

13-? 
18.3 
22.9 



27 

2.7 

3-1 
3-6 
4.6 

4-5 
9.0 

13-5 
18.0 

22.5 





19 


18 


6 


1-9 


1.8 


7 


2.2 


2.1 


8 


2.5 


2.4 


9 


2.8 


2.8 


10 


3-1 


.3.1 


20 


6.3 


6.1 


30 


9-5 


9.2 


40 


12.6 


12.3 


50 


15-8 


15.4 



I 

9 

10 

20 
30 
40 
50 



9 

0.9 
1.6 
1.2 
1-3 
1-5 
3-0 

4-5 
6.0 

7.5 



26 

2-6 
3-1 
3-S 
4.0 

4.4 

8.8 
13.2 

17-6 
22.1 



18 

1.8 
2.1 
2.4 
2.7 
30 
6.0 
9.0 
12.0 
15.0 



8 

0.8 
o 
I 

3 

4 



P. P. 



ai> 



382 



TABLE VII.-LOGARITHMIC SINES. COSINES. TANGENTS. AND COTANGENTS 



;}5 



Lop. Sill. 



9-75^59 
9-75877 
9.75895 

9-75913 
9-75 931 



9-75 949 
9.75967 
9.75985 
9.76003 
9.76021 



9.76039 
9.76057 
9.76075 
9.76092 
g.'jd no 



19 



9.76 128 

9.76146 

9.76 164 
9.76 182 
9. 'j6 200 



9.76 21^ 
9.76235 
9.76253 
9.76 271 
9.76 289 

9-76306 
9.76324 
9.76342 
9.76 360 
9-7637? 



9-76395 
9.76413 

9.76431 

9-76443 
9.76466 



9-76484 
9.76 501 
9.76519 

9-76536 
976554 



9.76572 

9765S9 
9.76607 
9 76 624 
9.76 642 



9.76 660 
9.76677 
9.76695 
9.76 712 
9.76730 



9-76747 
9.76765 
9.76 782 
9.76 800 
9.76 8if 



9-76835 
9.76 852 
9.76 869 
9.76887 
9.76904 



9.76 922 



18 
18 
18 
18 

18 

18 
18 
18 
18 
18 
18 
18 

If 
18 

18 

iS 

If 
18 
18 

If 
18 
18 

If 
18 

If 
18 

If 
18 

If 
18 

If 
18 

If 
If 
18 

If 
If 
If 
18 

If 
If 
If 
17 
18 

17 

If 

If 

If 

If 

If 

If 

If 

If 

If 

If 

17 

If 

If 

If 

If 



fiOe. Tiin. r. d. 



9 84522 

9-84549 
9-84576 
9. 84 603 

9.84630 



9.84657 
9.84684 
9.84 71 1 

984 73f 
9.84764 



9.8479! 
9.84818 
9.84845 
9.84871 
9.84898 



9.84925 
9.84952 
9.84979 
9.85 005 
9-85032 



9.85059 
9.85 086 

9-85 113 
985 139 
9.85 166 



9.85 193 
9.85 220 
9.85246 
9.85273 
9.85 300 



9.85327 

985353 
9.85 380 

9.85407 
9-85433 



9.85 466 
9-85487 

9-85513 
9.85 540 

9-85567 



594 



9.85 
9.85 620 
9.85647 
9-85673 
9.85 700 



9.85727 

9-85753 
9.85780 
9.85 807 
9.85833 



9.85 860 
9.85887 
9.85913 
9.85940 
9.85966 



Log. Cos. I d. 



985993 
9.86 020 

9.86046 
9.86073 
9. 86 099 



9.86 126 



27 
27 
27 
26 

27 
27 

27 

26 

27 
27 

26 

27 

26 

27 

27 

26 

27 

26 

27 

27 

26 

27 

26 

27 

26 

27 

26 

27 

26 

27 

26 
26 

27 

26 

27 

26 
26 

27 

2S 

27 

26 
26 
26 

27 

2S 

26 

27 

26 
2S 

26 

27 

26 
26 
26 
26 

27 

26 
26 
26 
26 



Lotr. Cot. 



o-i5 47f 

0.15456 
0.15423 

0-15 396 
o. 1 5 370 



0-15343 
O.I53I6 

o. 15 289 
o. 15 262 

0.15235 



0.15 208 

0.15 182 

0.15 155 

0.15 128 

0.15 loT 



o. 1 5 074 
o. 1 5 048 
o 1 5 02 1 
o. 14 994 
o. 14 967 



Loir. Cos. 



9-9' 336 
9.91 327 

9.91 318 
9.91 310 
9.91 301 

9.91 292 
9.91 283 
9.91 274 
9.91 265 
9-91 256 

9-91 24f 
9.91 239 
9.91 230 
9.91 221 
9.91 212 



9.91 203 
9.91 194 



o. 1 4 940 
0.14 914 
0.14887 
0.14 866 
0.14833 



9.91 

9-91 
9.91 



185 

176 
167 



0.14807 
0.14 780 

0-14753 
0.14726 
o. 14700 



o. 14 673 
0.14646 
o. 14620 

0.14593 
0.14566 



9.91 
9.91 

9.91 
9.91 
9.91 



158 
149 

146 

131 

122 



9.91 113 
9.91 104 
9-91 095 
9.91086 
9.91 077 



0.14539 
0.14513 

0.14486 
0.14459 

0-14 433 



o. 1 4 406 
0.14379 

o- 14 353 
0.14326 
o. 14 299 



9.91 068 
9.91 059 
9.91 056 
9.91 041 
9.91 032 



9.91 023 
9.91 014 
9.91 005 
9.90996 
9.90987 



0.14 273 
o. 14 246 
0.14 219 
0.14 193 
0.14 166 



0.14 140 
0.14 113 
o. 14 086 
o. 1 4 060 
0.14033 



9.90978 
990969 
9. 90 960 
9.90951 
9.90942 



990933 
9.90923 
9.90914 
9.90905 
9-90896 



Log. Cot. I c. (1. 



o. 14007 
o. 1 3 980 

0.13953 
0.13927 
o. 13 906 

Log. Tan. 



9.90887 
9.90878 
9. 90 869 
9. 90 860 
9.90 856 



9.90 841 
9-90832 
9-90823 
9.90814 
9.90 805 



9.90796 



9 

9 

8 

9 

9 

8 

9 

9 

9 

9 

8 

9 

9 

9 

9 

9 

8 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 



Log. Sill. 



59 

58 
57 
56 



55 
54 

53 
52 
51 



50 

49 

48 
47 
46 



45 

44 

43 
42 

41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



24 

23 
22 
21 

"20 

19 
18 

17 
16 



15 
14 

13 
12 

1 1 

To 

9 

8 

7 
_6 

5 
4 



1*. i' 



6 

7 
8 

9 
10 

20 

30 

40 
50 



27 

2.7 

3-1 
3-6 
4.6 

4-5 
9-0 

13 5 
18.0 
22.5 



28 



6 

7 
8 

9 
10 

20 
30 
40 
50 



18 

1.8 
2.1 
2.4 
2.7 
30 
6.0 
9-0 
12.0 
15.0 



If 

i-f 
2.6 

2-3 
2.6 
2.9 

5.8 

ir.6 
14.6 



17 

1-7 
2.0 
2.2 
2.5 
2-8 
5.6 
8.5 
II-3 
14. 1 



6 

7 
8 

9 
10 

20 

30 



9 

0.9 
1. 1 
1.2 
1.4 
1.6 

3-1 
4.f 



40 6.3 



9 

0.9 
1.6 
1.2 
1-3 
1-5 
3-0 
4-5 



8 

0.8 
i.o 
I.I 

13 
1-4 

2.8 
4.2 



6.0 5.6 



5oj7-9l7.5l7.i 



\\ V 



54 



383 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 



10 

II 

12 

14 



15 
16 

18 

19 



20 

21 
22 

23 

24 



25 
26 

27 
28 
29 



30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 

45 
46 

47 
48 

49 



9.76 922 
9.76939 

9-76950 
9.76974 
9.76991 



Log. Sill. I (I. 
17 

17 
If 
If 
17 
if 
17 
If 
If 
17 
If 



9.77 008 
9.77 026 

9-77043 
9.77 066 
9.77078 



9.77095 
9.77 112 
9.77 130 
9.77 147 
9.77 164 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



60 



9.77 181 

9-77 198 
9.77 216 

9-77233 
9.77250 



9.77 267 
9-77284 
9.77302 

9-77319 
9-77336 



9-77 353 
9-77370 
9-77387 
9.77404 
9.77421 



9-77 439 
9-77456 

9-77 473 
9.77490 

9-77 507 



9-77 524 
9-77 541 
9-77558 
9-77 575 
9-77 592 



9-77 609 
9.77 626 

9-77643 
9.77 660 

9-77 ^77 



9-77693 
977710 
9-77 72f 
9-77 744 
9.77761 



9-77778 

9-77 795 
9.77812 

9.77828 

9-77845 



9.77 862 

9-77879 
9.77896 

9-77913 
9.77929 



9-77 946 



Log. Cos. 



17 
17 
If 
17 
If 

17 
17 
If 
17 
17 
If 
17 
17 
17 
17 
If 
17 
17 
17 
17 



Log. Tan. 
9.86 126 
9.86 152 
9.86 179 
9.86 206 
9.86 232 



17 
17 
17 

17 
17 
17 
17 
17 

16 
17 
17 
17 
17 
16 
17 
17 
16 
17 
17 
16 
17 
17 
16 
17 



9.86 259 
9.86285 
9.86 312 

9-86338 
9-86365 



9.86 391 
9.86418 
9.86444 
9.86471 
9-86497 



(1. 



86524 
.86550 

.86577 
.86603 
86630 



9 



.86656 
9.86683 
9.86709 
9.86 736 
9.86762 

9-86788 
9.86815 
9.86841 
9.86868 
9.86894 



9.86 921 

9.86 94f 

9-86973 

9. 87 000 

9 -87026 



9.87053 
9.87079 
9.87 105 
9.87 132 
Q-87 158 
1.87 185 



9 



II 



9.872 
9.87237 
9.87 264 
9.87 290 



:56 



c. d. 



9-87 3I6 

987343 
9.87369 

9-87395 

9.87422 

"9^448 
9.87474 
9.87501 
9-87 52f 
9-87553 



9.87 580 
9.87 606 
9.87 632 

9-87659 
9.87685 

9-87 711 



Log. Cot. 



Log. Cot. 



26 
26 

27 

26 

26 
2§ 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 

26 
26 
26 
26 
26 
26 

26 

2§ 
26 
26 
26 

26 

26 
26 
26 

26 

26 
26 
26 

26 

26 
26 

26 

2S 
26 

26 

26 

26 

26 

26 
2§ 

26 

26 

26 

26 
cTd^ 



0.13874 
0.1384^ 

0.13 821 

0.13794 

o.i3 76f 



0.13 741 

O.I37I4 
0.13688 

0.13 661 

0.13635 



9-90796 
9.90786 

9-90 77f 
9.90 768 

9.90759 



o. 1 3 608 

0.13 582 

0.13555 
0.13529 

o. 1 3 502 



0.13476 
0.13449 
0.13423 

0.13396 
0.13 370 



0.13343 
O.I33I7 

0.13 290 
0.13 264 

0.13237 



9.90750 
9-90740 
990731 

9.90 722 

9.90713 



9.90703 
9-90694 

9.90685 
9.90676 

9.90666 



9.9065^ 

9. 90 648 

9.90639 

9.90 629 
9. 90 620 



0.13 211 

0.13 185 
0.13 158 

0.13 132 
0.13 105 



0.13079 

0.13 052 

0.13026 

o. 1 3 000 

0.12973 



o. 1 2 947 

O. I 2 920 

0.12 894 

0.12868 

0.12 841 



O. 12 815 

0.12 789 

O. 12 762 

0.12 736 

O. I 2 709 

0.12 683 
0.12 657 

O. 1 2 636 
0. 1 2 604 

0.12 578 



9.9061 1 
9. 90 602 

9.90592 
9.90583 

9-90 574 



9.90 564 
9-90 555 

9.90 546 

990 536 
9.90 527 



9.90 518 
9-90 508 
9-90499 
9-90490 
9.90 486 



9.90471 

9.90 461 

9.90452 

9-90443 
9-90433 



0.12 551 
0.12 525 
0.12 499 
0.12 472 
0.1 2 446 



0.12 420 
0.12 393 
o. 12 36f 
0.12 341 
0.12315 



9.90424 
9.90414 

9-90405 
9.90396 

9-90386 



9.90377 

9-90 36f 
9.90358 

9-90348 
9-90339 



9-90330 
9-90 320 
9.90 311 

9-90301 
9.90 292 



o. 12 288 



Log. Tan. 



9.90 282 
9-90 273 
9.90 263 
9.90254 
9.90244 



990235 



Log. Sin. 



Log. Cos. I d. 



9 
9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 



d. 



00 

59 
58 

57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 

43 
42 

41 
40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



25 

24 

23 
22 
21 



20 

19 
18 

17 
16 



15 
14 

13 
12 

II 



10 

9 
8 

7 
6 



p. P. 



6 

7 
8 

9 
10 
20 
30 
40 
50 



6 

7 
8 

9 
10 
20 
30 
40 
50 



27 28 



2.7 


2.6 


3-1 


3-1 


3-6 


3-5 


4.0 


4.0 


4-5 


4-4 


9.0 


8.8 


13-5 


13-2 


18.0 


17-6 


22. ^ 


22.1 



26 

2.6 

3-0 

3-4 
3-9 
4-3 
8.6 
13.0 

17.3 
21. § 



If 17 



I.f 


1.7 


2.0 


2.0 


2-3 


2.2 


2.6 


2.5 


2.9 


2.8 


5-8 


5-6 


8.7 


8.5 


II. 6 


II. J 


14.6 


14.1 



18 

1.6 

1-9 

2.2 

2.5 

2.f 

5-5 
8.2 

II. o 

i3-f 



6 

7 
8 

9 
10 
20 

30 

40 

50 



9 

0.9 
I.I 
1.2 
1.4 
1.6 

3-1 
4-7 
6.3 
7-9 



9 

0.9 
i.o 
1.2 
1-3 
1-5 
3-0 

4-5 
6.0 

7.5 



P. P. 



53° 



3^4 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 






' ! Loer. sill. 



9 

10 

1 1 

12 



13 
14 



15 
16 

18 
19 



20 

21 
22 

23 

24 



25 
26 

27 
28 

29 
30 

31 
32 
33 
34 



J3 
36 

37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



m 



9-77 946 
9.77963 
9.77980 

9-77 996 
9.78013 



9.78030 
9.78046 
9 78063 
9.78080 
9.78097 



9.78 113 
9.78 130 
9.78 147 
9.78 163 
9.78 180 



9.78 196 
9.78213 
9.78 230 
9.78246 
9.78263 



9.78 279 
9.78 296 
9.78 312 
9.78329 
9.78346 



9.78 362 

9-78379 
9-7839? 
9.78412 
9.78428 



9.78444 
9.78461 

9-78 47f 
9.78494 
9.78510 



9.78 527 
9-78 543 

978559 
9.78576 
9-78 592 



9.78 609 
9.78625 
9.78 641 
9.78658 
9.78674 



9.78 696 
9.78707 
9.78723 
9-78739 
9-78755 



9.78772 

9-78788 
9.78 804 
9.78821 
9.78837 



9.78853 
9.78869 
9.78885 
9.78 902 
9.78918 



9- 78 934 



liOtr. (;os. I 



liOer. Tan. r. «l. 



9-87711 

9-8773? 
9.87764 
9.87796 
9-87 816 



9-87843 
9.87869 
9.87895 
9.87 921 
9-87948 



9.87974 
9.88 000 
9.88 026 

9.88053 

9.88 079 



9.88 105 

988 I3T 

9.88157 

9.88 184 

9.88 210 



9.88236 
9.88262 
9.88288 

9.88315 
9.88341 



9.88 367 

9.88393 

9.88 419 

9-88445 

9.88472 



9.88498 
9.88 524 
9.88556 

9-88 576 
9.88602 



9.88629 
9.88655 
9.88681 
9.88707 
988733 



9.88759 
9 88785 
9.88 81T 
9.88838 
9.8 8864 

9. 88 890 
9.88916 
9.88942 
9.88968 
9-88 994 



9.89 026 
9.89046 
9.89 072 
9.89098 
9.89 124 



9.89 156 

9-89 ^11 
9.89203 
9.89 229 

9-89255 



9.89 281 



26 
26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 
26 

26 

26 

26 

26 

26 

26 
26 

26 

26 
26 

26 

26 
26 

26 

26 
26 

26 

26 
26 

26 

26 
26 
26 

2S 

26 
26 
26 

26 

26 
26 
26 
26 
26 

26 

26 
26 
26 
26 
26 
26 

26 

26 
26 
26 
26 



l.op. Cot. 



o. 



IjOu. (!()t. c. d. I Loir. Tan 



2288 
2 262 
2 236 
2 209 
2183 



2 157 
2 131 
2 104 

2 078 

2 052 



2 026 

999 
973 
947 
921 



895 
868 
842 
816 
790 



763 
IZl 
711 
685 
659 



371 
345 
319 
293 
266 



240 
214 
188 
162 
136 



1 10 
084 
058 
032 
005 



0979 

0953 
092^ 
o 90T 
0875 



o 849 
o 823 

0797 
0771 
0745 



07 '9 



Dtf. Con. I d. 



90235 
90 225 
90 216 
90 206 

90196 



90 187 

90177 

90 168 

90158 
90 149 



90 139 
90 130 

90 120 
90 116 

90 lOI 



90091 
90082 
90072 
90062 

90053 



90 043 
90033 

90024 
90014 
90004 



89995 

89985 
89975 

89966 

89 956 



89946 
89937 
89927 
8991? 
89908 



89898 
89888 
89878 
89869 
89859 



89 849 
89839 
89 830 
89820 
89816 

89791 
89781 
89771 
89761 



89751 
89742 
89 732 
89722 
89712 



89702 
89692 
89683 
89673 
89663 



989653 



9 

9 

9 

10 

9 

9 

9 

9 

9 

9 

9 
10 

9 

9 

9 

9 

lO 

9 
9 

9 
10 

9 

9 

10 

9 

9 
10 

9 

9 

10 

9 

9 

10 

9 
10 

9 
10 

9 
10 

9 
10 

9 
10 

9 
10 

9 
10 

10 

9 
10 

9 
10 

10 

9 
10 
10 

9 
10 

10 
10 



Lot;. Sill. 



<>0 

59 
58 
57 

55 
54 
53 
52 
51 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 

23 
22 
21 

"20" 

19 
18 

17 
16 



r. \\ 



2§ 26 



6 

7 
8 

9 
10 

20 

30 
40 

50 



2-6 


2 


3-1 


3 


3-5 


3- 


4.0 


3- 


4.4 


4- 


8.8 


8. 


13.2 


13- 


17-6 


17- 


22.1 


21. 





17 


18 


6 


1-7 


1-6 


7 


2.0 


1.9 


8 


2.2 


2.2 


9 


2-5 


2-5 


10 


2-8 


2-y 


20 


5-6 


5-5 


30 


8.5 


8.2 


40 


11-3 


II. 


50 


14.1 


13.7 



16 

1.6 

1-8 

2.1 
2.4 
2-6 

5-3 
8.0 

10.6 

13-3 





10 


6 


I.O 


7 


I.I 


8 


1-3 


9 


1-5 


10 


1-6 


20 


3-3 


30 


5.0 


40 


6 6 


50 


8-3 



9 

0.9 
1. 1 
1.2 
1.4 
1.6 

3-1 
4-? 
6.3 
7.9 



5:^ 



385 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

88^ 



10 

II 

12 

14 



15 
16 

18 
19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 

43 
44 



60 



Lo^. Sin. 



9-78934 
9.78 950 

9.78965 
9.78 982 
9.78 999 



9.79015 
9.79031 
9.79047 
9.79063 
9.79079 



9.79095 
9.79 III 
9.79 12^ 

9-79 143 
9.79 159 



9.79175 
9.79 191 
9.7920^ 
9.79223 
9.79239 



9-79255 
9.79271 

9.79287 

9-79303 

9 79319 



9-79 33^ 
979351 
979367 
979383 
9-79 399 



9.79415 

9-79431 
9-79 446 
9.79462 

9-79 478 



9-79 494 
9.79510 
9.79526 
9-79541 
9-79 557 



9-79 573 
9-79589 
9.79605 
9.79 620 
9-79635 



9.79652 
9.79668 
9.79683 

9 79 699 
979715 



9.79730 

9-79 746 
9.79762 
9.79777 
9-79 793 



9.79809 
979824 
9.79840 
9.79856 
9.79871 



9-79887 



Log.J^os^ 



d. 



Loff. Tan. 



9.89 281 
9.89307 

9-89333 

9-89359 
9.89385 



9.89 411 

9-89437 
9.89463 
9.89489 
9.89515 



9-89 541 
9.89 567 

9-89 593 
9.89 619 
9.89645 



9.89 671 
9.89697 
9.89723 

9-89749 
9.89775 



9.89 801 
9.89827 
9.89853 
9.89879 
9.89905 



9.89931 

9-89957 
9.89 982 

9.90008 
9.90034 



9.90066 
9.90085 
9. 90 112 

990 138 
9.90 164 



9.90 190 
9.90 216 
9.90 242 
9 90 268 
9.90294 



9.90319 

9-9c» 345 
9.90371 
9.9039; 
9.90423 



9.90449 

990475 
9.90 501 

9.90 525 

9:90552 



990578 
9. 90 604 

9.90630 
9.90656 
9.90 682 



9.9070; 

9-90733 
9.90759 
9.90785 
9.90 81 1 



9.90837 



Log. Cot. 



c. d. 



26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

25 
26 
26 
26 
26 
26 
26 
2$ 
26 
26 
26 
26 
26 

25 
26 
26 
26 

25 
26 
26 
26 

25 
26 

26 
26 

2? 
26 
26 

2? 
26 
26 
26 

25 
26 



c. d. 



Log. Cot. 



0719 
0693 
0667 
0641 
0615 



0589 
0563 

0537 
o 511 

0485 



0459 

0433 
0407 

0381 
0355 



0329 
0303 

o 277 
o 251 

o 225 



o 199 
o 173 

0147 

O 121 
0095 



o 069 

0043 

o 01; 
0.09 991 
0.09 965 



0.09939 
0.09913 
0.09 887 
0.09 861 
0.09 836 



0.09 810 
0.09 784 
0.09 758 

0.09 732 

0.09 706 



0.09 686 
0.09 654 
0.09 623 
0.09 602 
0.09 577 



0.09551 
0.09 525 
0.09 499 
o 09 473 
0.09 44; 



0.09 421 
0.09 395 
0.09 370 
0.09 344 
0.09 318 



0.09 292 
0.09 265 
0.09 246 
0.09 214 
0.09 189 



0.09 163 



Log. Tan. 



Los. Cos. 



9.89653 
9.89643 

9-89633 
9.89623 
9.89613 



9.89 604 
9-89594 
9-89584 
9.89574 
9.89564 



9.89554 
9-89544 
9-89534 
9.89524 
9.89514 



9.89 504 

9-89494 
9.89484 
9.89474 
9.89464 



9-89454 
9-89444 
9-89434 
9.89424 
9.89414 



9.89404 
9.89394 
9-89384 
989374 
9.89364 



9-89354 
9.89344 

9-89334 
9.89324 
9.89314 



9.89304 
9.89294 
9.89 284 

9-89274 
9.89 264 



9.89253 
9-89243 
9-89233 
9.89223 
9.89213 



9.89 203 
9.89193 
9.89 182 
9.89 172 
9.89 162 



9.89152 
9.89 142 
9.89132 
9.89 I2T 
9 89 1 1 1 



9.89 lOI 
9.89091 
9.89081 
9.89076 
9.89066 



9.89 056 



Log. Sin. 



d. 



GO 

59 

58 
57 
56 



55 
54 
53 
52 
51 
50 

49 
48 

47 

45 
44 
43 
42 
41 



40 

39 
38 
V 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



15 
14 

13 
12 
II 



10 

9 
8 

7 

6 



p. P. 





26 


2S ; 


6 


2.6 


2.5 s 


7 


3-0 


3.0 


8 


3-4 


3-4 


9 


3-9 


3-8 


10 


4-3 


4.2 ; 


20 


8.6 


8-5 ; 


30 


13.0 


12.; i 


40 


17.3 


17.0 


50 


21.6 


21.2 1 





16 


16 


6 


1-6 


1.6 


7 


1.9 


1-8 


8 


2.2 


2. 1 


9 


2-5 


2.4 


10 


2.7 


2-6 


20 


5-5 


5-3 


30 


8.2 


8.0 


40 


II. 


10.6 


50 


13-y 


13-3 



15 

1.8 
2.6 

2-3 
26 

5-1 

1-1 

i,o-3 

12.9 





10 


10 


6 


1.6 


I.O 


7 


1.2 


I.I 


8 


1-4 


I-.3 


9 


1.6 


i.S 


10 


1.7 


1-6 


20 


3-5 


3-3 


30 


5-2 


5.0 


40 


7-0 


6.5 


50 


8.; 


8.3 



9 

0.9 
I.I 
1.2 

1.4 
1.6 

3-1 

^•1 

6.3 
7.9 



p. p 



51° 



386 



TABLE VII. — LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 



10 

1 1 

12 



15 

16 

17 
18 

19 



20 

21 
22 

23 
24 



25 
26 

27 

28 

29 



80 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 

42 
43 
44 



45 
46 

47 
43 

50 

51 
52 
53 
54 



Lojf. Sill. I (1. Loi?. Tan. 



55 
56 
57 
58 
59 



60 



9.79887 
9.79903 
9-79918 
9-79 934 
9-79 949 



9.79965 
9.79980 
9.79996 
9.8001T 
9.80027 



9.80042 
9.80058 
9.80073 
9.80089 
9.80 104 



9.80 120 
9.80135 
9. 80 1 5 1 
9.80 165 
9 80 182 



9.80 197 
9. 80 2 1 3 
9.80228 
9.80243 
9.80 259 



9.80 274 
9.80 289 
9.80305 
9.80 320 
9-80335 



980351 
9.80366 
9.80381 
9.80397 
9 80412 



9 80427 
9.80443 
9.80458 
9 80473 
9 8048Q 



9.80 504 
9.80 519 
9.80534 

9-80549 
g.8o 564 

9.80 580 
9.80595 
9. 80 610 
9. 80 62 1 
9. 80 646 

9.80655 
9.80671 
9.80686 
9.80 701 
9-80 716 



9-80731 
9.80746 
9.80 761 

9-80776 
9.80791 



). 80 806 



16 

15 
15 

15 
15 
15 
IS 
15 
iS 
15 
15 
15 
il 
il 
15 
IS 
iS 
iS 
15 
15 
15 
15 
15 
15 
15 
iS 
15 
15 
15 
15 
15 
15 
15 
15 
15 
15 
15 
15 
15 
IS 
15 
15 

IS 
15 
IS 
15 
15 
15 
IS 
15 
15 
IS 
15 
15 
15 
15 
15 
15 



jiOg. Cos . i < 1. 



90837 
90863 
90 8S3 
90914 
90940 



90 966 
90 992 
01^ 

043 
069 



095 
121 

i4o 
172 

198 



224 
250 

27S 

301 

327 



353 
378 
404: 

430 

456 



481 
507 
533 

559 
584 



616 
636 
662 
68^ 
713 



739 
765 
790 
816 
842 



867 

893 
919 

945 
970 



996 
92 022 
92 04f 
92073 
92 099 



92 124 
92 150 
92 176 
92 201 
92 22^ 



92253 

92 278 
92304 
92 330 

9235S 



92 381 



26 

2S 
26 

25 
26 
26 

2S 
26 
26 

2S 

26 

2S 
26 

2S 
26 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

2S 
26 

25 

2S 
26 

2S 
26 

2S 

2S 
26 

2S 
26 

2S 

2S 
26 

2S 
2S 
26 

2S 

2S 
26 

2S 

2S 
26 



L(«r. Cot- 



0.09 163 
0.09 137 

0.09 1 1 T 
0.09085 
0.09 060 



0.09034 
0.09008 
0.08 982 
0.08956 
0.08 930 



0.08 905 
0.08 879 
0.08853 
0.08827 
0.08 802 



0.08 776 
0.08 750 
0.08 724 
0.08 693 
0.08 673 



0.08 647 
0.08 621 
0.08 595 
0.08 570 
0.08 544 



0.08 518 
0.08 492 
0.08 467 
0.08 441 
0.08 415 



0.08 389 
0.08 364 
0.08 338 
0.08 312 
0.08 286 



0.08 261 
0.08 235 
0.08 209 
0.08 183 
0.08 158 



0.08 132 
0.08 106 
0.08 081 
0.08 055 
0.08 029 

0.08 004 
0.07 978 
0.07 952 
0.07 926 
0.07 901 

0.07 875 
0.07 849 
0.07 824 
0.07 798 
0.07 772 



0.07 747 
0.07 721 
0.07 695 
0.07 670 
0.07 644 



o. 07 6 1 8 



liOp. Cot. 1 c. «1. I liO^. Tan. 



Lour. Cos. 



(I. 



89056 
89040 
89030 
89019 
89009 



88 999 
88989 
88978 
88968 
88958 



88947 

88937 
88927 
88917 
88906 



88 896 
88 886 
88875 
88865 
88855 



88844 
88834 
88823 
88813 
88803 



88792 
88782 
88772 
88761 
88751 



88746 
88730 
88720 
88 709 
88699 



88 688 
88678 
88667 
88657 
88646 



88636 
88625 
88615 
88604 
88594 



88 583 

88573 
88562 
88 552 
88 541 



88531 
88 526 
88 510 
88 499 
88489 



9-88 478 
9.88467 

9.88457 
9.88446 
9-88 436 



9.88425 



liOi?. Sill. 



10 
10 
10 
10 
16 
10 
16 
16 
10 

i5 
10 
10 
10 
16 
16 
10 
16 
16 
10 
i5 
16 
16 
10 
16 
16 
16 
10 
16 
16 

10 
10 
10 
16 
16 
16 
16 
16 
i5 
16 
10 
16 
16 
10 
10 
16 
i5 
16 
16 
10 
i5 
10 
10 
10 
i5 
10 
I r 
16 
16 
10 
10 



50 

49 
48 

47 
j^ 

45 
44 

43 
42 

41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



25 

24 

23 
22 

21 

19 
18 

17 
16 

~^ 
14 

13 
12 

1 1 

lo 

9 
8 

7 
6 



1'. r. 



26 2S 



7 
8 

9 

10 

20 

30 
40 

50 



2.6 


2.S 


3-0 


3-0 


3-4 
3-9 


3-4 
3-8 


4-3 
8.6 


4-2 
8-5 


13.0 


12.5^ 


17.3 
21.5 


17.0 
21.2 



6 

7 
8 

9 
10 

20 

30 
40 

SO 



16 

1.6 

1-8 
2.1 

2.4 
2-6 

5-3 
8.0 

10.6 

13-3 



IS 

i-S 

1.8 
2.6 

2.3 
2.6 

5-1 

7-f 

10.3 

12.9 



15 

1-5 
I-? 
2.0 
2.2 

2-5 
5-0 

7.5 
1 0.0 
12.5 





II 


10 


I 


6 


I.I 


1.6 




7 


1-3 


1.2 




8 


1.4 


1.4 




9 


1-6 


1.6 




10 


1-8 


1-7 




20 


3.6 


3-5 


3- 


30 


5-5 


5-2 


5- 


40 


7-. 3 


7-0 


6. 


50 


9.1 


8.? 


8. 



.0 
.1 
■3 
.5 
•6 

■3 
.0 

-6 

•3 



p. I'. 



50 



337 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

40° 



10 

II 

12 

13 

14 



15 
i6 

17 
i8 

19 



20 

21 
22 

23 

24 



25 
26 
27 
28 
29 



35 
36 
37 
38 
39 



40 

41 
42 

43 

44 



45 
46 

47 
48 

49 



Loe. Sill. 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
(JO 



9.80805 
9.80822 
9.80837 
9.80852 
9.80867 



9.80882 
9.80897 
9.80 912 
9.80927 
9.80942 



9.80957 
9,80972 
9.80987 
ooT 
016 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



ao 


9.8 


31 


9.8 


32 


9.8 


33 


9.8 


34 


9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 

9^ 
9.8 



031 

046 
061 

076 
091 



106 
121 
136 
150 
165 



186 

195 
210 

225 
239 



254 
269 
284 
299 
313 



328 
343 
358 
372 
387 



402 
416 

431 
446 
460 



475 
490 
504 

519 

534 



548 
563 
578 
592 
607 



621 
636 
650 
665 
680 

694 

Log. Cos. 



(1. 



15 
15 
15 
15 
15 
15 
15 
15 
15 
15 
15 
14 
15 
15 
15 
15 
15 
14 

15 
15 
15 
14 
15 
15 
14 
15 
15 
14 

15 
14 
15 
15 

15 

14 

15 

14 

14 

15 

14 

15 

14 

14 

15 

14 

14 

15 

14 

1$ 

14 

15 

14 
14^ 
14^ 
14 
15 
14 



Lop. Tan. 



9.92 38T 
9.92407 
9.92432 

9-92 458 
9.92484 



9.92 509 

9-92 535 
9.92 561 
9.92 585 
9.92 612 



9.92638 
9.92663 
9.92 689 
9.92714 
9.92 746 



9.92 766 
9.92 791 
9.92 817 
9.92 842 
9.92 868 



9.92894 
9.92919 
9.92945 
9.92971 
9.92996 



C. (1. 



9.93022 

9-9304? 
993073 

9-93 098 
9.93 124 



9.93 150 

9-93 175 
9.93 201 

9.93225 
9.93 252 



9-93 278 
9-93 303 
993329 
9-93 354 
9-93380 



9-93405 
9-93 431 
9-93 456 
9.93 482 

9-93 508 



9-93 533 
9-93 559 
9-93 584 
9.93610 

9-93635 



9.93661 
9-93685 
9.93712 

9-93 73? 
9-93763 



9-93788 
9.93814 
9.93 840 
9.93 865 
9-93891 

9-93 9^6 
Log. Cot. 



25 
25 
26 

25 

25 

25 
26 

25 
25 
26 

25 
25 
25 
26 

25 
25 
25 
25 
26 

25 
25 
25 
26 

25 
25 
25 
25 
25 
26 

25 
25 
25 
25 
25 
26 

25 
25 
25 
25 
25 
25 
25 
25 
26 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 
26 

25 

25 

25 

cTdT 



Log. Cot. 



0.07 618 
0.07 593 
0.07 567 
0.07 54T 
0.07 516 



0.07 490 
0.07 465 
0.07 439 
0.07 413 
0.07 388 



Log. Cos. 



9.88425 
9.88415 
9.88404 

9-88393 
9-88383 



(1. 



9.88372 
9.88 36T 
9.88351 
9.88346 
9.88 329 



0.07 362 

0.07 336 
0.07 311 
0.07 285 
0.07 259 



0.07 234 
0.07 208 
0.07 183 
0.07 15^ 
0.07 1 31 



0.07 106 
0.07 086 
0.07 055 
0.07 029 
0.07 003 

0.06 978 
0.06 952 
0.06 927 
0.06 901 
0.06 875 



0.06 850 
0.06 824 
0.06 799 
0.06 773 
0.06 748 



0.06 722 
0.06 695 
0.06 671 
0.06 645 
0.06 620 



0.06 594 
0.06 569 
0.06 543 
0.06 518 
0.06 492 



0.06 465 
0.06 441 
0.06415 
0.06 390 
0.06 364 



9.88319 
9-88308 
9.8829^ 
9.88 287 
988275 
9.88265 
9.88255 
9.88 244 
9-88233 
9.88223 



9.88 212 
9.88 201 
9.88 196 
9.88 180 
9.88 169 



158 

147 

137 
126 

115 



9.88 104 
9. 88 094 
9.88083 
9.88072 
9.88 061 



9.88050 
9.88039 
9.88029 
9.88018 
9.88007 



9.87995 
9.87985 
9-87974 
9.87963 

9-87953 



0.06 339 
0.06 313 
0.06 288 
0.06 262 
0.06 237 



0.06 21 T 
0.06 186 
0.06 160 
0.06 134 
0.06 109 

0.06083 

Log. Tan. 



9.87942 
9.87931 
9.87 920 
9.87909 
9.87898 



9.8788? 

9-87876 
9.87865 

9-87854 
9.87844 



9-87833 
9.87 822 

9.87 811 
9.87 800 
987789 
9-87778 
liOg. Sin. 



10 

II 

16 

16 

16 

II 

10 

16 

II 

id 

16 

1 1 

16 

16 

1 1 

16 

id 

II 

16 

1 1 

16 

II 

16 

II 

16 

II 

16 

II 

16 

II 

id 

II 

II 

16 

II 

II 

16 

II 

II 

16 

II 

II 

II 

16 

II 

II 

II 

16 

II 

II 

II 

II 

II 

16 

II 
II 
II 
II 
II 
II 



GO 

59 
58 

57 
56 



p. p 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 

43 
42 

41 



40 

39 
38 
37 
36 



35 
34 

33 

12 

31 



30 

29 
28 

27 
26 



25 
24 

23 
22 
21 

To" 

19 
18 

17 
16 



15 
14 

13 
12 
1 1 



10 

9 

8 

7 
6 



26 2$ 



6 

7 
8 

9 
10 

20 

30 
40 

50 



2.6 


2.5 


30 


3-0 


3-4 


3-4 


3-9 
4-3 
8.6 


3-8 

4.2 
8.5 


13.0 


12.^ 


17-3 
21.6 


17.0 
21.2 





15 


S5 


6 


1-5 


1-5 


7 


1.8 


I.? 


8 


2.0 


2.0 


9 


2.3 


2.2 


10 


2.6 


2.5 


20 


5-1 


5.0 


30 


7.? 


7.5 


40 


10.3 


lO.O 


50 


12.9 


12.5 



14 

1.4 

1-7 
1.9 

2.2 

2.4 

4-8 
7.2 

9-6 
12. 1 





II 


10 


6 


I.I 


I.O 


7 


1-3 


1.2 


8 


1.4 


1.4 


9 


1-6 


1.6 


10 


1-8 


1-7 


20 


36 


3.5 


30 


5-5 


5-2 


40 


7-3 


7-0 


50 


9.1 


8.? 



P. p. 



49 



388 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, AND COTANGENTS. 



11 



5 
6 

7 
8 

9 
10 

II 

12 

13 
14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



Lotr. sill. 



25 
26 

27 
28 

29 

ao 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

\^ 
43 
44 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 
9.8 
9.8 
9.8 
9.8 



9.8 



694 
709 

723 
738 

752 



767 
781 
796 
810 
824 



839 

853 
868 
882 
897 



911 
925 
940 

954 

q6o 



983 
997 



9.82 012 
9.82 025 
9.82 040 



9.82055 
9.82 069 
9.82 083 
9.82 098 
9 82 112 



9.82 126 
9.82 146 
982 155 
9.82 169 
9.82183 



9.82 197 
9.82 212 
9.82 226 
9.82 246 

9.82 2U 



9.82 269 
9.82283 
9.82 297 
9-82311 
9.8232^ 



45 
46 

47 
48 

49 



:>o 

51 

52 

53 
54 



55 
56 
57 
58 
59 

m 



9-82339 
9.82354 
9.82368 
9.82 382 

9-82 396 



9.82 410 
9.82424 
9.82438 
9.82452 
9.82467 



9.82 481 
9.82495 
9.82 509 
9.82 523 
9-82537 
9-82 551 

Lot?. Cos. »K 



liOur. Tan. c. «l 



9939I6 

9 93942 
9.93967 

9-93 993 
9.94018 



9.94044 
9.94069 
9.94095 
9.94 120 
9.94146 



9-94171 
9.94 197 
9.94 222 
9.94248 
9-94273 



9-94299 

9.94324 

9.94350 

9-94 375 
9.94400 



9.94426 
9.94451 

9-94 477 
9.94502 

9-94528 



9-94 553 
9-94 579 
9.94604 
9.94630 
9-94655 



9.94681 

9-94 706 
9-94732 
9-94 757 
9-94782 

9. 94 808 
9-94833 

9-94859 
9.94884 
9.94910 



9-94 935 
9.94961 

9.94986 

9.95011 

9-95037 



9.95 062 
9.95 o83 

9-95 113 
9-95 139 
9.95 164 



9.95 189 

9-95 215 
9.95 240 
9.95 266 
9-95291 



9-95 316 
9-95 342 
9-95367 
9-95 393 
9-95 418 

9-95 443 

liOtr. ("ot. c. (I. 



25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
2S 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
2^ 
25 
25 
2? 

25 
25 
25 
25 
25 

25 



lA.ir. Cot. 


Lou'. Cos. (1. 




0.06 083 


9-87778 ,, 


<><) 


0.06 058 


9 


87 767 ; ; 


59 


0.06 032 


9 


87756 




58 


0.06 007 


9 


87745 




57 


0.05 981 


9 


87734 




56 

55 


0.05 956 


9 


87723 


0.05 930 


9 


87712 




54 


0.05 905 


9 


87701 




53 


0.05 879 


9 


87 690 




52 


0.05 854 


9 


87679 




51 
oO 


0.05 828 


9 


87668 


0.05 803 


9 


87657 




49 


0.05 777 


9 


87645 




48 


0.05752 


9 


87634 




47 


0.05 726 


9 


87623 




46 


0.05 701 


9 


87612 


45 


0.05 67I 


9 


87 601 




44 


0.05 650 


9 


87 590 




43 


0.05 625 


9 


87579 




42 


0.05 599 


9 


87 568 




41 


0.05 574 


9 


87557 


40 


0.05 548 


9 


87546 




39 


0.05 523 


9 


87535 


1 


38 


0.05 497 


9 


87523 




37 


0.05 472 


9 


87512 


^ '■ 


36 


0.05 446 


9 


87 501 


35 


0.05 421 


9 


87490 




34 


0.05 395 


9 


87479 




33 


0.05 370 


9 


87468 




32 


0.05 344 


9 


87457 




31 


0.05 319 


9 


87445 


30 


0.05 293 


9 


87434 


^ 


29 


0.05 268 


9 


87423 




28 


0.05 243 


9 


87412 




27 


0.05 217 


9 


87 401 


j^ 


26 


0.05 192 


9 


87389 


25 


0.05 166 


9 


87 378 


^ 


24 


0.05 141 


9 


87367 




23 


0.05 III 


9 


87356 




22 


0.05 090 


9 


87345 




21 


0.05 064 


9 


87333 


20 


0.05 039 


9 


87322 


•^ ^ 


19 


0.05 014 


9 


8731^ 




18 


0.04 988 


9 


87300 




17 


0.04 963 


9 


.87 2(88 


j| 


16 


0.04 937 


9 


.87277 


15 


0,04 912 


9 


87266 


^ 


14 


0.04 886 


9 


87254 




13 


0.04 861 


9 


87243 


^ 


12 


0.04 836 


9 


87232 




II 
10 


0.04 816 


9 


87 221 


0.04785 


9 


.87 209 


^ 


9 


0.04759 


9 


87 198 




8 


0.04734 


9 


87187 




7 


0.04 708 


9 


87175 




6 

5 


0.04683 


9 


87164 


0.04658 


9 


87153 


^ 


4 


0.04632 


9 


87 141 


^ 


3 


0.04607 


9 


87 130 


^ 


T 


0.04 581 


9 


87 118 




I 


0.04 556 


9 


87 107 


\a)S. Tan. 


1 


our. Sin. 


(1. 


t 



V. V 



2S 25 



6 

7 
8 

9 
10 
20 
30 
40 
50 



2 


5 


--> 


3 





2. 


3 


4 


3- 


3 


8 


3- 


4 


2 


4- 


8 


5 


8. 


12 


1 


12. 


17 





16. 


21 


2 


20. 





14 


6 


1.4 


7 


1.7 


8 


1-9 


9 


2.2 


10 


2.4 


20 


4-8 


30 


7-2 


40 


9.6 


50 


12. 1 



14 

1-4 

1.6 

1-8 
2.1 

2-3 
4-6 
7-0 

9-3 
II. 6 





II 


I 


6 


I.I 




7 


1-3 




8 


1-5 




9 


1-7 




10 


1.9 




20 


3-8 


3- 


30 


5-7 


5- 


40 


7-6 


7- 


50 


9-6 


9- 



I'. V 



48 



3»9 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

42" 



10 

II 

12 

13 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 

24 



26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



50 

51 
52 

53 
54 

55 
56 
57 
58 
59 

(io 



Loff. Siu. 



82551 
82565 
82 579 

82593 
82 607 



82621 
82635 
82649 
82663 
82 677 



82691 
82 705 
82 719 
82733 
82746 



82 766 

82774 
82788 
82802 
82816 



82830 
82844 
82858 
8287T 
82885 



82899 
82 913 
82 927 
82 940 
82954 



82968 
82982 
82 996 
83009 
83023 



83037 
83051 
83 064 

83078 
83092 



83 106 
83 !I9 

83 133 
83 147 
83 166 



83 174 
83 188 
83 201 

83215 
83229 



83 242 
83256 
83 269 
83283 
83297 



,10 



83 

83324 

83337 

83351 

83365 

83 378 



Log. Cos. (1. 



Lofr. Tan. c. <1. I Loer. Cot. 



9-95 443 
9.95 469 

9.95494 
9.95 520 

9-95 54^ 



9-95 571 
9.95 596 
9.95 621 
9.95647 
9.95672 



9.95697 
9.95723 
9-95 748 
9-95 774 
9-95 799 



9.95 824 
9-95850 

9-95873 
9.95901 
9.95 926 



95951 

95 977 

96 002 
96027 
96053 



9.96078 
9.96 104 
9.96 129 
9.96154 
9.96 180 



9.96 205 
9.96230 
9.96 256 
9.96 281 
996306 



9-96332 

9-96357 

9-96383 
9. 96 408 

9-96433 



9-96459 
9.96484 

9-96 509 

9-96535 
9.96 560 



9-96 58S 
9.96 611 
9.96636 
9.96 661 
9.96687 



9.96 712 
9.96737 
9.96763 

9-96788 
9-96813 

9.96839 
9.96 864 
9.96889 
9.96915 
9.96 940 

9-9696! 

Lot,'. Cot. c. d 



25 
25 

2l 

A 
25 
25 
2! 
25 

2S 
25 
25 
25 
25 
25 

25 
25 
25 
2^ 
25 
25 
25 
25 
25 
2? 

2S 
25 
25 
2? 
2? 
25 
25 
25 
25 
25 
25 
25 
25 
25 
2! 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
2? 
25 
25 
25 
25 
25 
25 

25 



0.04 556 
0.04531 
0.04 505 
0.04 480 
0.04454 



0.04429 
0.04404 
0.04 378 
0.04353 
0.04327 



0.04 302 
0.04 277 
0.04 251 
0.04 226 
0.04 206 



0.04175 
0.04 I 50 
0.04 124 
o. 04 099 
0.04074 



0.04048 
0.04 023 
0.03 997 
0.03 972 
0.03 947 



Loe. Cos. 



0.03 921 
0.03 896 
0.03 871 
0.03 84! 
0.03 820 

0.03 795 
0.03 769 
0.03 744 
0.03 718 
0.03693 



0.03 668 
0.03 642 
0.03 617 
0.03 592 
0.03 565 



0.03 541 
0.03 516 
0.03496 
0.03465 
0.03 440 



0.03414 
0.03 389 
0.03 364 
0-03 338 
0.03313 



0.03 28^ 
0.03 262 
0.03 237 
0.03 21 T 
0.03 186 



0.03 161 
0.03 135 
0.03 1 16 
0.03 085 
0.03059 

0-03034 

Log. Tan. 



9.87 107 
9.87 096 
9.87 084 
9-87073 
9.87 062 



9.87 056 
9.87039 
9.87 027 
9.87 016 
9.87 004 



9-86993 
9.86982 
9.86 976 
9.86959 
9-86947 



9-86936 
9.86 924 
9-86913 
9.86 90T 
9.86890 



9-86 878 
9.86867 
9.86855 
9.86 844 
9.86832 



9.86821 
9.86809 
9.86798 
9.86786 
9-867 74 
9.86 763 
9.86751 
9.86 740 
9.86728 
986 716 



9.86 705 
9.86 693 
9.86682 
9.86 676 
9-86658 



9.86647 
9.86 63I 
9.86 623 
9.86612 
9. 86 606 



86588 

86577 
86565 

86553 
86542 

9.86 530 

9-86 518 
9.86 507 
9.86495 
9.86483 



9.86471 
9. 86 460 
9.86448 
9.86436 
9.86424 

9.86412 

Los. Sin. 



00 

59 

58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



24 

23 

22 
21 



20 

19 
18 

17 
16 



15 
14 

13 
12 

II 



10 

9 



.5 
4 
3 
2 
I 



p. P. 



25 



6 

7 
8 

9 

10 

20 
30 
40 
50 



2.S 


2. 


3-0 


2. 


3-4 


3- 


3-8 


3- 


4.2 


4- 


8.5 


8. 


12.7 


12. 


17.0 


16. 


21.2 


20. 



25 

5 
9 

3 
7 
I 

3 
5 
6 





14 


6 


1.4 


7 


1-6 


8 


1.8 


9 


. 2.1 


10 


2-3 


20 


4-6 


30 


7.0 


40 


9-3 


50 


II. 6 



13 

1-3 
1.6 
1.8 
2.0 

2.2 

4-5 

6.7 

9.0 

II. 2 





12 


II 


I] 


6 


1.2 


I.I 




7 


1.4 


1.3 




8 


1.6 


1.5 




9 


1.8 


1.7 




10 


2.0 


1.9 




20 


4.0 


3-8 


3- 


30 


6.0 


5-7 


5- 


40 


8.0 


7-6 


7. 


50 


lO.O 


9.6 


9- 



p. p. 



47' 



390 



TABLE VIL— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

43° 



10 

II 

12 

13 

14 

15 
16 

17 
18 

19 

20 

21 

22 

23 
24 

25 
26 

27 
28 

29 

30 

31 

32 
33 
34 

35 
36 
37 
38 
39 
40 

41 

42 

43 
44 

45 
46 

47 
48 

49 
50 

51 

52 
53 
54 

55 
56 
57 
58 
59 
60 



Lofj. Sin. (1. 



83378 
83392 
83405 
83419 
83432 



83446 
83459 
83473 
83486 
83500 



83513 
83527 
83540 
83554 
83567 



83585 

83594 
83607 
83621 
83634 



83647 
83661 

83674 
83688 
83701 



83714 
83728 

83741 
83754 
83768 



83781 

83794 
83808 
83821 
83834 



83847 
83861 

83874 
83887 

83900 



83914 
83927 
83940 

83953 
83967 



83980 

83993 
84005 

84019 
84033 



84046 

84059 
84072 
8408I 
84098 



84 III 
84 124 
84138 
84 151 
84 164 

84177 



Log. Cos. (I. 



Loff. Tnii. c. (1 



9.96965 
9.96991 
9.97 016 

9-97041 
9.97067 



9.97092 
9.97 117 

9-97 143 
9.97 168 

9-97 193 



9.97219 
9.97 244 
9.97 269 
9.97 295 
9.97320 



9-97 345 
9-97 370 
9.97396 

9-97421 
9-97 446 



9.97472 

9-97 497 
9-97 522 
9 97 548 
9-97 573 



9-97 598 
9.97624 
9.97649 

9-97 674 
9-97699 



9.97725 
9.97 750 

9-97 775 
9.97 801 
9.97 826 



9-97 851 
9-97 877 
9-97902 
9.9792^ 
9.97952 



9-97 978 
9.98 003 
9.98028 
9-98054 
9.98079 



9.98 104 
9.98 129 
9.98155 
9.98 186 
9.98 205 



9-98231 
9.98 256 
9.98 281 
9-98 306 
9-98 332 



9-98 357 
9.98 382 
9.98 408 
998433 
9-984 58 
9-98483 

Lop. Cot. 



25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 



Lop. Cot. 



0.03034 
0.03 009 
0.02 984 

0.02 958 
0.02933 



0.02 908 
0.02 882 
0.02 857 
0.02 832 
0.02 806 



0.02 781 
0.02 756 
0.02 736 
0.02 705 
0.02 680 



0.02 654 
0.02 629 
0,02 604 
0.02 578 
0.02 553 



0.02 528 
0.02 502 
0.02 47^ 
0.02 452 
0.02 427 



0.02 401 
0.02 376 
0.02 351 
0.02 325 
0.02 306 



0.02 275 
0.02 249 
0.02 224 
0.02 199 
0.02 174 



0.02 148 
0.02 123 
0.02 098 
0.02 072 
0.02 04^ 



0.02 022 
0.0 1 996 
o.oi 971 
0.0 1 946 
0.01 921 



0.01 895 
0.01 876 
0.01 845 
0.01 819 
0.01 794 



0.01 769 
0.01 744 
0.01 718 
0.01 693 
0.01 668 



0.01 642 
0.01 61^ 
0.01 592 
0.01 567 
0.01 541 

o-oi 5 16 

Lor. Tan. 



Loe. Cos. 



9.86354 
9.86342 
9.86330 
9-86318 
9- 8630 6 



9.86 294 
9.86282 
9.86 271 
9.86 259 
9.86 247 



86412 
86 401 
86389 
8637? 
86365 



86235 
86 223 
86 21 1 
86 199 
86187 



86 176 
86164 
86 152 
86 140 
86128 



86 116 
86 104 
86092 
86080 
86068 



86056 
^ 86044 
9.86032 
9.86 020 
- 86 008 



85996 

85984 
85972 
85 960 
85948 



85936 
85924 
85 912 
85 900 
85887 

85875 
85863 
85851 

85839 
85827 



85815 
85803 
85791 

85 778 
85766 



85754 
85742 
85730 
85718 

85705 
985693 

Lot;. Sin. 



(io 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 

48 

47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



25 

24 

23 
22 
21 



20 

19 
18 

17 
16 



I'. V. 





2S 


25 


6 


2-5 


2.5 


7 


3-0 


2.9 


8 


3-4 


3.3 


9 


3-8 


3.^ 


10 


4.2 


4.1 


20 


8.5 


8.3 


30 


12. f 


12.5 


40 


17.0 


16.6 


50 


21.2 


20.8 





13 


13 


6 


1-3 


1.3 


7 


1.6 


1.5 


8 


1.8 


i.^ 


9 


2.0 


1.9 


10 


2.2 


2.1 


20 


4-5 


4.3 


30 


6.7 


6.5 


40 


9.0 


8.S 


50 


II. 2 


I0.8 





12 


12 


II 


6 


1.2 


1.2 


I.I 


7 


1-4 


1.4 


1-3 


8 


1-6 


1.6 


1-5 


9 


1-9 


1.8 


1-7 


10 


2.1 


2.0 


1-9 


20 


4.T 


4.0 


3-8 


30 


6.2 


6.0 


S-7 


40 


8.3 


8.0 


7-6 


50 


10.4 


lO.Q 


9.6 



1'. F. 



46 



391 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

44° 



10 

II 

12 

14 



15 

i6 

i8 
19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 

54 



55 
56 
57 
58 
59 
60 



Log. Sin. 



9.84177 
9.84 190 
9.84 203 
9.84 216 
9.84229 



9.84 242 

9.84255 

9.84268 
9.84281 
9.84294 



9.84307 
9.84320 

9-84333 

9-84 346 
9-84359 



9.84372 
9-84385 

9-84398 
9.84411 
9.84424 



9-84437 
9.84450 
9.84463 
9.84476 
9.84489 



9.84502 
9.84514 
9.84 52f 
9.84540 
9-84553 



9.84 566 

9-84579 
9-84592 
9. 84 604 
9.8461^ 



9. 84 630 
9.84643 
9.84656 
9.84669 
9.84681 



d. 



9.84694 
9.84707 
9.84720 
9.84732 
9.84745 



9.84758 

9-84771 

9-84783 

9-84796 
9.84809 



9.84822 

9-84834 
9.84847 
9.84860 
9.84872 



9.84885 
9.84898 
9.84 916 
9.84923 
9-84936 

9-84948 
Lo g. Cos. 



13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
12 

13 
13 
13 
13 

13 
12 

13 
13 
13 
12 

13 
13 
12 

13 

13 
12 

13 
13 
12 

13 
12 

13 

12 

13 
12 

13 
12 

13 
12 

13 
12 
12 

13 
12 

12 

13 
12 
12 

13 
12 



Log. Tan. 



9.98483 
9.98 509 
9-98 534 
9-98 559 
9.98 585 



9.98 610 
9.98635 
9. 98 666 
9.98686 
9.98 711 



9-98736 
9.98 762 

9-98787 
9.98 812 

9.98837 



9.98863 
9.98888 
9.98913 

9-98 938 
9.98964 



9.98989 
9.99014 
9.99040 
9.99065 
9.99096 



9-99115 
9.99 141 
9.99 166 
9.99 1 91 
9-99 216 



9.99242 
9.99267 
9.99292 
9.99318 

9-99 343 



9-99368 
9-99 393 
9.99419 

9-99 444 
9.99469 



9.99494 
9-99 520 
9-99 545 
9.99570 

9-99 59? 



9.99 621 
9.99646 
9.99671 
9.99697 
9.99722 



9-99 74^ 
9-99772 
9.99798 
9-99823 
9-99848 



c. d. 



9-99873 

9-99899 
9.99924 

9-99 949 
9-99 974 
0.00000 



2S 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

2§ 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 



Log. Cot. 



o.oi 516 
o.oi 491 
O.OI 465 
O.OI 446 
O.OI 415 



O.OI 390 
O.OI 364 
O.OI 339 
O.OI 314 
O.OI 289 



O.OI 263 
O.OI 238 
O.OI 213 
O.OI 18^ 
O.OI 162 



O.OI 137 

O.OI 112 
O.OI 086 
O.OI 061 
O.OI 036 



O.OI 010 
0.00 985 
0.00 960 
0.00 935 
0.00 909 



0.00 884 
0.00 859 
0.00 834 
0.00 808 
0.00 783 



0.00758 

0.00733 

0.00 70^ 
0.00 682 
0.00 657 



0.00 631 
0.00 606 
0.00 581 
0.00 556 
0.00 536 



Log. Co t, led. 



0.00 505 
0.00 480 
0.00455 
0.00429 
0.00404 



0.00 379 
0.00353 
0.00 328 
0.00 303 
0.00 278 



0.00 252 
0.00 227 
0.00 202 
0.00 177 
0.00 151 



0.00 126 
0.00 lOI 
0.00076 
0.00056 
0.00025 
o. 00 000 
Lo g. Ta n. 



Log. Cos. 



9.85693 
9.85681 
9.85 669 
9.85657 
9.85644 



9.85 632 
9.85 620 
9.85608 

9-85 595 
9-85 583 



9.85571 
9.85559 
9-85 546 
9-85 534 
9.85 522 



9-85 509 
9-8549^ 
9.85485 
9-85472 
9.85 466 



9.85448 

9-85435 
9.85423 

9.85411 
9-85 398 



9.85 386 

9-85374 
9.85361 

9-85349 
9-85336 



9.85 324 
9.85312 

9-85299 
9.85 287 

9-85 274 
9,85 262 

9-85 249 
9.85 237 
9.85 224 
9.85 212 



d. 



9.85 199 

9-85 187 
9.85 174 
9.85 162 
9.85 149 



9-85 137 
9.85 124 
9.85 112 
9.85099 
9.85087 



9.85074 
9.85 062 
9.85049 
9.85037 
9.85 024 



9.85 Oil 

9.84999 
9.84986 
9.84974 
9.84961 

9-84948 
Log. aiu. 



d. 



GO 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 

43 
42 

41 



40 

39 
38 

37 
36 



25 
24 

23 

22 

21 



20 

19 
18 

17 
16 



15 

14 

13 
12 
II 



10 

9 
8 

7 
6 



P. P. 



6 

7 
8 

9 
10 

20 

30 
40 

50 



6 

7 
8 

9 
10 
20 

30 
40 

50 



2S 25 



2-5 


2. 


3-0 


2. 


3-4 
3-8 


3. 
3- 


4.2 
8.5 


4- 
8. 


I2.f 


12. 


17.0 


16. 


21.2 


20. 



12 

1.2 

1-4 
1-6 
1-9 
2.1 
4.1 
6.2 

8-3 
10.4 





13 


13 


6 


1-3 


1.3 


7 


1.6 


i.S 


8 


1.8 


i-f 


9 


2.0 


1.9 


10 


2.2 


2.1 


20 


4.5 


4.3 


30 


6-^ 


6.5 


40 


9.0 


8.6 


50 


II. 2 


10.8 



12 

1.2 

1.4 

1.6 
1.8 
2.0 
4.0 
6.0 
8.0 

lO.O 



p. p. 



45 



392 



TABLE VIII. 

LOGARITHMIC VERSED SINES AND EXTERNAL 

SECANTS. 



TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

0° 1° 




15 
16 

17 
18 

19 



20 

21 

22 

23 

24 



25 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 
43 

44 



45 
46 

47 
48 

49 



Los. Vers. 



2> 



50 

51 
52 
53 

54 



55 
56 
57 
58 

i9. 
60 



CO 



2.62642 
3.22848 
3.58066 
3-83054 



4.02436 
.18272 
.31662 
.43260 
• 53490 



4.62642 
,70920 

.78478 
.85431 
.91- ■ 



4.97860 

5.03466 

.08732 

•13696 
•18393 



5.22848 
. 27086 
•31126 
• 34987 
.38684 



5.42230 
•45636 

.48915 
.52073 
.55121 



5 . 58066 
.60914 
.63672 
. 66344 
•68937 



5^7i455 
.73902 
.76282 

•78598 
.80854 



5-83053 
•85198 
.87291 

•89335 
.91332 



5.93284 

•95193 

.97061 

5.98890 

6 . 00680 



6.02435 

.04155 
.05842 

•07496 
.09120 



6. 10714 
. 12279 
.13816 

.15327 
.16811 

6.18271 

Log. Vers. 



Loe. Exsec. 



60206 

352I8 

24987 

19382 

15836 

13389 
I I 598 

10230 
915T 

8278 
7558 
6953 
6437 

5992 
5605 
5266 
4964 
4696 

4455 
4238 
4046 
3861 
3697 

3545 
3406 
3278 
3158 
3048 

2944 
2848 
2757 
2672 

2593 
2518 
2447 
2379 
23'6 
2256 

2199 
2145 
2093 
2044 
1996 
1952 
1909 
1868 
1829 
1790 

1755 
1720 



1654 
1623 

1594 
1565 

1537 
1511 

1484 
1460 

J) 



— 00 
2,62642 
3.22848 
3.58066 
3-83054 



4^02436 
, 18272 
,31662 
,43260 
•53491 



4.62642 
.70921 
.78478 
•85431 
.91868 



4.97861 

5^03466 
.08732 

•13697 
•18393 



5.22849 
.27087 
.31127 

• 34988 
.38685 



■5.42231 

•45638 
.48916 

•52075 
•55123 



. 58068 
.60916 

•63674 
•66346 
. 68940 



5^71457 
.73904 
.76284 
.78601 
.8085^ 



5-83056 
.85201 
.87295 
.89338 
-91335 



5-93288 

•95197 

.97065 

5.98894 

6.00685 



6 . 02440 
.04160 
.05847 
.07501 
.09125 



n 



6. 10719 
. 12284 
. 13822 

.15333 
.16818 

6.18278 

Los. Exsec. 



Log. Vers. 



J» 



60206 
35218 
2498? 
19382 
15836 
13389 

1 1 598 
10236 

9151 
8279 

7557 
6952 
6437 

5993 
5605 
5266 
4964 
4696 

4456 
4238 
4046 
386T 
3697 

3545 
3407 
3278 

31591 
3048 

2945 
2848 

2758 
2672 

2593 

251? 
2447 
2380 

2316 

2256 

2199 
2145 

2093 

2043 
1997 

1952 
1909 

1829 

I79I 

1755 

1720 

1687 
1654 
1623 

1594 
1565 

1537 
I5II 

1485 

1460 



7> 



6,18271 
.19707 
.211 19 
.22509 

.23877 



6.25223 
.26549 
.27856 
.29142 
.30416 



6. 31666 
,32892 
.34107 

•35305 
.36487 



6.37653 
• 38803 

. 39938 
,41059 
.42165 



43258 
.44337 
.45403 
.46455 
. 47496 



6.48524 

.49539 
.50544 
•51536 
.52518 



6.53488 
. 54448 
.55397 
•56336 
.57265 



6.58184 

• 59093 

• 59993 
.60884 

.61766 



6.62639 
.63503 
•64359 
.65206 
.66045 



6.66876 
,67700 
.68515 

.69323 
.70124 



6.70917 
.71703 
.72482 
.73254 
.74019 



6.7477^ 

.75529 
.76275 

.77014 

.777 aJ 
6.78474 

Loir. Vers. 



Log. Exsec. 



1435 
I412 

1389 
1368 

1346 
1326 

1306 
1286 
1268 

1250 
1232 
1214 
1 1 98 
1182 
1 166 
1 1 50 

1135 
1121 
1 106 

1093 
I078 
1066 
1052 
1046 

1028 

1015 

1004 

992 

981 

970 
960 

949 

939 
929 

919 

909 
900 
891 
882 
872 
864 

855 
84^ 

839 
831 
823 

815 



806 

793 
786 

779 
772 
765 

758 
752 

745 
739 
733 
726 

/> 



6.18278 
.19714 

.21126 
.225I6 
.23884 



6.2523T 
.2655^ 
.27864 
,29151 
•30419 



6.31669 
,32901 
-34II6 
•35315 
• 36497 



6.37663 
.38814 

• 39949 
.41076 

.42177 



6.43270 

•44349 

•45415 
•46468 
•47509 



48537 
49553 
■50557 
■51550 

■52532 



6^53503 
• 54463 
•55413 
•56352 
•57281 



. 58201 
.59116 
.60011 
. 60902 

.61784 



6.62657 
,63522 
.64378 
.65226 
.66065 



6.66897 

,67726 

■68536 

•69345 
.70145 



6.70939 
.71725 
.72505 
.7327^ 
. 74043 



6.74802 

•75554 
, 76306 
. 77040 
.7777% 
6.78506 

I, Off. Kxst'C 



2) 



1436 
I412 
1390 
1368 

1347 
1326 

I3O6 
1287 
I26g 

1250 
1232 
I215 

II98 
1182 

1 166 
1151 

1135 
1121 

1106 
1093 
1079 
1066 

1053 
1046 

1028 
1016 
1004 

993 

982 

976 
960 

950 

939 
929 

919 
909 
906 
891 

882 

873 
864 
856 
848 
839 

83I 
823 
816 
808 
806 

794 
786 

779 
772 
765 

759 
752 

746 

739 

733 
727 



10 

II 
12 

13 

14 



15 
16 

17 

18 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 



30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 

44 



4S 
46 
47 
48 
49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 
60 



394 



TABLE VIII.- 



-LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

2° :r 



5 
6 

7 
8 

9 
10 

II 

12 

14 



15 

i6 

17 
i8 

19 



20 

21 



24 



25 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 

49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



(>0 



Lou:. A'ers. 



/> 



78474 

79195 
79909 

8061 8 
81322 



82019 
,82711 
,83398 
, 84079 
•84755 



85425 
86091 

8675t 
,87407 
,88057 



88703 

89344 
89980 
90612 
91239 



91862 
,92480 

93093 
93703 
94308 



94909 

95506 
, 96099 
,96688 
,97272 



•97853 
•98430 
• 99004 

•99573 
•00139 



.00701 
.01259 
.01814 
.02366 
.02914 



03458 
.03999 
.04537 
.05071 
.05603 



. 06 1 30 
.06655 
.07177 
.07695 
.0821 1 



•08723 
.09232 

•09739 
. 10242 

• 10743 



1 1240 

11735 
1 222^ 

12716 
13203 



13687 



7_ 

IjOsj. »rs. 



721 

71-+ 
709 
703 
697 
692 
686 
681 
676 
670 
665 
666 

655 
656 

646 
641 
636 
631 
627 
622 
618 
613 
609 
605 

601 
597 
592 
589 

58-+ 
581 

577 
573 
569 

565 
562 
558 
555 

551 

548 

544 
541 
537 
534 
531 
527 

525 

521 

518 
515 
512 

509 

506 

503 
506 

497 
495 
492 
489 
486 
484 



IjO!?. Kxsec 



T> 



6.78500 
.79221 

• 79937 
. 80645 
.81350 



6.82048 
.82740 

.83427 
.84109 

•84785 

6.85457 

.86123 

.86783 

•87439 
. 88096 



6.88737 

•89378 
.90015 
.90647 
.91275 



6.91898 
.92516 
•93131 
•93741 
• 94346 



6 . 94948 

•95545 
.96139 
.96728 
•973J3 



6.97895 
.98472 

•99046 
6.99616 
7.00182 



7.00745 
.01304 
.01860 
.02412 
.02966 



7.03505 
.04047 

•04585 
.05126 
.05652 



7 . 06 1 86 
. 06706 
.07228 

■0774? 
.08263 



7.08776 
.09286 

•09793 
. 10297 

• I0798 



1 1297 
1 1792 
12285 

12775 
13262 



7> \ Loif. Vers. 



/> 



7 • 1 3746 



721 

715 
709 

703 
698 
692 
687 
682 
676 
671 
666 
666 
656 
651 

646 
641 

636 
632 

628 

623 

61 8 
614 
610 
605 

601 

597 
593 
589 
585 
581 
577 
574 
570 
566 

563 
559 

555 
552 
548 

545 
541 
538 
535 
531 
528 
525 

r '>2 

519 
516 

513 
509 

507 

503 
501 

498 
495 
493 
490 
487 
484 



Kxser. 



/> i 



7 



13687 
14168 
14646 
15122 

15595 



16066 

16534 
17000 

17463 
17923 



18382 
18837 
19291 
19742 
201 91 



20637 
21081 
21523 
21963 
22406 



22836 
23269 
23700 
24129 

24555 



24980 
25402 
25823 
2624T 
26658 



27072 
27485 
27895 
28304 
287 II 



29116 

29518 
29919 

30319 
307 1 6 



31112 
31505 
3189^ 
32288 

32676 



33063 
33448 
33831 
34213 
34593 



34971 
35348 
35723 
36097 
36468 

36839 
37207 

37574 
37940 
38304 
38667 



481 
478 
475 
473 
470 
468 
466 

463 
466 

458 
455 
453 
451 
448 
446 
444 
442 
440 

437 

435 
433 
431 
429 

426 

424 
422 
426 
418 
4>6 
414 
412 
416 
409 
406 

405 
402 
401 

399 
397 

395 
393 
392 
390 
388 
386 
385 
383 
382 

380 

378 
377 
375 
373 
371 
370 
368 
367 
366 

364 
^62 



l,(n:. K\s<'c 



liOer. Vers. 



7> 



13746 
14228 
14707 

15183 
15657 



1 61 29 
16598 
17064 
17528 
17989 



18448 
18905 

19359 
1 98 II 
20260 



2070^ 
21 152 
21595 
22035 
22473 



22909 
23343 
23775 
24204 
24632 



25057 
25486 
25902 
26321 
26738 



27153 
27567 
27978 
28387 

28795 



29200 
29604 
30006 

30406 
30804 



3 1 201 

31595 
31988 
32379 
32768 



33 '56 
33542 
33926 

34309 
34689 



35069 

35446 
35822 

36196 

36569_ 

36940 

373'0 
37678 

38044 
38409 



It 



38773 



481 
479 
476 
474 

471 

469 
466 
464 
46 1 

459 
456 
454 
452 
449 
447 
445 
442 
446 

438 

436 

434 

431 
429 

427 
425 
423 
421 
419 
41^ 

415 
413 
411 

409 
407 

405 
404 
402 
400 
398 

396 
394 
393 
391 
389 
388 

385 
38-; 
382 
386 

379 
377 
376 
374 

37' 

569 

368 

366 
365 
363 



liOe. Kxscr. 



/> 



395 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

4° 5° 







10 

II 

12 

13 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 



30 

31 
32 

33 
34 



35 
36 
37 
38 
39 



Lof?. Vers. 



40 

41 

42 
43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



7.38667 
. 39028 

•39387 

.39745 
.40102 



7.40457 
.40810 
.41163 

.41513 
.41863 



D Loff. Exsec. 2> 



.4221 I 

•4255^ 
.42903 

.43246 
.43589 



7.43930 
.44270 

.44608 
.44946 
.45281 



7.45616 

.4594-9 
.46281 
. 466 1 2 
.46941 



7.47270 

.47597 
.47922 
.48247 
.48570 



7.48892 
.49213 
.49533 
.49852 
.50169 



7. 5048 S 
. 50800 

.51114 

.51427 
. .51739 



7 



52050 

52359 
5266^ 

.52975 
.53281 



7.53586 
. 53890 
.54193 
. 54495 
. 54796 



60 



7.55096 

.55395 
.55692 

.55989 
.5628^ 



7.56580 
.56873 
.57166 
•57458 
. 57749 



7.58039 



361 

359 
358 
356 

355 
353 
352 
350 
349 
348 
346 
345 
343 
342 

341 
339 
338 
337 
335 
334 

332 
330 
329 

328 
327 
325 
324 
323 
322 
321 
320 
318 
3^7 
316 

315 
314 
313 
311 

311 
309 
308 
30^ 
306 

305 
304 

303 
302 

300 

300 

299 

29? 
297 
295 

295 
293 
293 
292 
290 
290 



Lo:r. Vers. 



7-38773 
•39134 
.39495 
.39854 
. 402 1 1 



7.4056; 
.40922 

.41275 
.41627 

.41977 



7.42326 
.42673 
.43019 
.43364 
.43708 



7.44050 
.44390 
.44730 
.45068 

.4540=; 



7.457401 
.46075 , 
.4640^ i 

.46739 i 
.47070 i 



7.47399 
.47727 
.48054 

.48379 
.48703 



7.49026 

.49348 
. 49669 

.49989 
.5030; 



7 . 50624 

. 50941 
.51256 

.51569 
.51882 



7.52194 

.52504 
.52814 
.53122 

.53429 



7.53735 
.54041 
. 54345 
. 54648 
.54950 



7 



n 



.55251 
.55550 
.55849 
.5614^ 

. 56444 



7.56740 
•57035 
•57329 
.57621 

•57913 



7.58204 



361 
366 

359 
35^ 
356 
354 
353 
352 
350 

349 
34^ 
346 
345 
343 
342 
340 
339 
338 
337 

335 
334 
332 
332 
330 

329 

328 

327 
325 
324 

323 
322 
321 

319 

318 

317 

316 

315 

313 

313 

311 
316 

309 

308 

307 

306 

305 

304 

303 
302 

301 
299 
299 
298 
296 
296 
295 

294 
292 
292 
291 



Log. Vers. 



7 



Los;. Kxsec 



7> 



7 



58039 
58328 

58615 
58902 

59188 



59473 
59758 
60041 
60323 
60604 



60885 
61 164 

61443 
61721 
61998 



62274 
62549 
62823 

63096 
63369 



63641 
63911 
6418T 
64451 
64719 



64986 

65253 
65519 
65784 
66048 



6631 1 
66574 
66836 
67097 
6735^ 



67617 

67875 
68133 
68396 
68647 



D Loff. Exsec. I 2> 



68902 
6915; 
6941 1 
69665 
6991; 



70169 
70421 
70671 
70921 
71 170 



714I8 
71666 

71913 
72159 
72404 



72649 
72893 
73137 

73379 
73621 



73863 



Lo!.'. Vers. 



289 
287 
287 
286 

285 
284 
283 
282 
281 
286 
279 
279 
27? 
277 
276 
275 
274 

273 

272 

272 
276 
270 
269 
268 

26^ 
266 
266 
265 
264 

263 
263 
26T 
261 
266 

259 

258 
258 

257 
256 

255 

255 
254 

253 
252 

252 
251 
250 
250 
249 

248 
24^ 
247 
246 
245 

245 
244 

243 

242 
242 
241 



7. 



58204 
58494 

58783 
59071 

59358 



59645 

59930 

,60214 

. 60498 

.60786 



7. 



7> 



61062 
61342 
61622 
6190T 
62179 



.62456 
.62733 
, 63008 
.63282 
.63556 



63829 
,64101 
,64372 
. 64643 
.64912 



,65181 
.65449 
.65716 
.65982 
.6624^ 



.66512 
.66776 
.67039 
.67301 
,67562 



,67823 
,68083 
,68342 
,68601 

,68858 



,69115 
,69371 
,69627 
.69881 
.70135 



70388 
,70641 
70893 
,71144 
71394 



71644 
,71892 
,72141 

•72388 
.72635 



,72881 
.73126 
.73371 
.73615 
•73859 



7-74iot 

Lour. Kxsec 



290 
289 
288 
287 

286 
285 
284 
283 
282 

281 
286 
280 
279 
278 

277 

276 
275 
274 
274 

273 
272 

271 
276 
269 

269 
268 
267 
266 
265 

264 
264 
263 
262 
261 
261 
260 

259 
258 
25; 

257 
256 

255 
254 
254 

253 

252 

252 
251 
250 

250 

248 
248 
24? 
246 

246 
245 
245 
244 

243 
242 



]> 





I 

2 

3 
4 

5 
6 

7 
8 

_9L 
10 

II 
12 

13 
14 



p. P. 



15 
]6 

17 
18 

19 



20 

21 

22 

23 

24 



25 
26 

27 
28 

29 



30 

31 

32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 

43 
44 



45 
46 
47 
48 
49 



50 

51 

52 
53 
54 

55 
56 
57 
58 
59 



6 


360 

36.0 


35.0 


7 
8 


42.0 
48.0 


40.8 
46.6 


9 
10 


54-0 
60.0 


51.5 
58., S 


20 


120.0 


116. 6 


30 


180.0 


175-0 


40 


240.0 


233.3 


50 


300.0 


291.6 



GO 



330 

33^o 
38.5 
44.0 

49.5 
55-0 

IIO.O 

165.0 

220.0 

275.0 



270 

27.0 

31-5 
36.0 

40-5 
45.0 

go.o 

135-0 
180.0 
225.0 



320 

32.0 
37-3 
42-6 
48.0 

53.3 
106.6 
160.0 
213.3 
266.6 



!6 

30.3 
34-6 
39-0 
43.3 
8^.-6 
130.0 

173.3 
216.6 



340 

34-0 
39-6 
45-3 
51.0 

56.6 
"3-3 
170.0 
226.6 
283.3 



310 

31.0 

36.! 

41-3 

46-5 

51-6 

103.3 

155-0 

206.6 

258.3 



300 290 280 



6 


30.0 


29.0 


7 


35.0 


33-8 


8 


40.0 


38-6 


9 


45 -o 


43.5 


10 


50.0 


48.3 


20 


100.0 


96.6 


30 


150.0 


145.0 


40 


200.0 


193-3 


50 


250.0 


241.6 



28. 

32-6 

37-3 
42.0 

46.6 

93-3 

140. 

1&6.6 

233 .3 



260 250 





240 


230 


6 


24.0 


23.0 


7 


28 .0 


26.8 


8 


32.0 


30.6 


9 


36.0 


34.5 


10 


40.0 


38.3 


20 


80.0 


76-6 


30 


120.0 


115.0 


40 


160.0 


153-3 


50 


200.0 


191-6 





210 


200 


6 


21.0 


20.0 


7 
8 


24-5 

28.0 


23.3 
26.6 


9 


31-5 


30.0 


10 
20 


35.0 
70.0 


33-3 
66.6 


30 


105.0 


100.0 


40 
50 


140.0 
175.0 


133-3 
166.6 



25- 
29. 

33-3 

37-5 

41-6 

83-3 

125.0 

166.6 

208.3 



220 

22.0 
25-6 
29-3 
33-0 
36.6 
73-3 

IIO.O 

146.6 
^83.3 



190 

19. 



25 

28, 

31 

63 

95 • 

126 



I', t*. 



396 



TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



G 



7° 



10 

1 1 

12 



14 



15 

i6 

1 8 
19 



20 

21 

2 2 

23 
ii_ 

23 
26 
27 
28 
29 
30 

3i 
32 
33 

34 



35 
36 
37 
i 38 
39 



40 

41 
42 

43 

44 

45 
46 
47 
4B 
49 



50 



54 



56 
57 
5S 
59 
(>0 



Loir. Vers. I J> 



7 



74104 
74344 

74583 
74822 



75060 
75297 
75534 
75770 
76006 



76246 

76475 
76708 

76941 

77173 



77405 

77636 
77867 
78097 
78326 



78554 
78783 
79010 

79237 
79463 



79689 I 

79914! 
80F38 
S0362 
805S6 



80808 
81031 
81252 

81473 
81694 



81914 
82133 
82352 
82570 
82788 



83005 
83222 
83438 

83653 
83868 



84083 

84297 
84516 

84723 
84933 



85147 

85359 
85570 
85780 
85990 



86199 
86408 
86616 
86824 
87031 

87238" 

Lou'. Vers. 



241 

240 
239 
239 

238 
237 

236 
236 
235 
234 
234 
233 
233 



231 

236 

230 
229 



2 ''7 
227 
226 
22 S 
225 
224 
22 1 
223 



221 
221 
226 

220 
219 
219 
218 
217 

217 
217 
216 
215 
215 
214 
214 
213 
213 
212 

212 

211 
21 I 
210 
210 
209 

209 
208 
208 
207 
206 

It 



Locr. Kxsec.I I> 



74101 

74343 
74585 
74826 
75066 



75305 

75544 
75782 
76019 
76256 



76492 
76728 

76963 
77197 

77431 



77664 
77S97 
78128 
78360 
78596 



78826 
79050 

79279 
79507 
79735 



79962 
80188 
80414 
80639 
80864 



81088 
81312 

8i53d 
81758 
81980 



82201 
82422 
82642 
82862 
83081 



83300 
83518 

83735 
83952 
84169 



84385 
84606 

84815 
85030 

85243 



85457 
85670 
85882 
86094 
86305 



86516 
86726 

86936 
87146 

87354 

87563 

liOir. Kxser. 



242 
24? 

241 

240 

239 
239 
238 

237 
237 

236 

235 
235 
234 
233 
233 

232 

231 
231 

230 

230 
229 
229 

228 
228 

227 
226 
226 
225 
225 
224 
224 
223 
222 
222 

221 
-> -> [ 

226 

I 219 

I 219 

' 219 

218 

217 

217 

3,6 

216 
215 
215 
214 
I 213 

I :: 
i 213 

' -'3 
212 

21T 

211 

21 1 

210 
210 
209 
208 
208 

7> 



Locr. Vers. J> Lok- Kxse( 



7.87238 

87444 
87650 
87855 
88060 



88264 
88468 
88672 
88875 
8907 f 



89279 
89481 
89682 
89882 
90082 



90282 

9048 T 
90680 

90878 
91076 



91273 
91476 
91667 
91863 
9205^ 



92253 

92448 
92642 

92836 
9 1029 



Q-3 2 22 

93415 
93607 

93799 
93990 



94181 
94371 
94561 

94751 
94940 



95129 
95317 
95505 
95693 
95880 



96066 
96253 

96439 
96624 

96809 



96994 

97178 
97362 
97546 
97729 



97912 
98094 
98276 
98458 
98639 
7 .08820 

lAta. Vt'rs. 



200 
205 
205 
204 

204 
204 
203 
203 
202 

202 
20 f 
201 
206 
200 

99 
99 
98 
98 
97 

97 
97 
96 
96 

95 

95 
95 
94 
94 
93 
93 
92 
92 
91 
91 

90 

96 

90 

89 
89 

89 
88 
87 
88 

87 

86 
86 
86 

85 

85 

84 

84 

84 

83 

83 

83 

82 

82 

82 

81 

81 



87563 
87771 

87978 
88185 
88391 



88597 
88803 
89008 
8921 2 
894 1 6 



89620 
89823 
90025 
90228 
90429 



90636 
90831 
91032 
91231 

91431 



It 



91630 
91828 
92027 
92224 
9242T 



926 1 8 
92815 
93016 
93206 
93401 



93596 
93790 
93984 
94177 
94370 



94562 
94754 
94946 
9513? 
95328 



95519 
95709 
95898 
96088 

96276 



96465 
96653 
96841 
97028 
97215 



97401 

9758? 
97773 
97958 
98143 



9832? 
98512 

98695 
98879 
99062 

90244 

/> I.Ki;. Ixsic. 



208 
207 
207 
206 

206 
205 

205 

204 
204 

203 

203 

202 

202 
201 

201 
201 

2C6 

99 
99 

99 
98 
98 
97 
97 
97 
96 
95 
95 
95 

95 
94 
94 
93 
93 
92 
92 
92 
91 
91 
96 
90 

89 
89 

88 

88 
88 
88 
87 
87 

86 
86 

85 
85 
84 
84 

84 
83 
83 
83 
82 



5 
6 

7 
8 

9 
10 

1 1 
12 

13 
14 



16 

17 
18 

19 



20 

21 

22 

23 
24 



r. I* 



J5 
36 

57 
38 
39 



40 

41 
42 
43 

44 

45 
46 

47 
48 

_49 

50 

51 
52 

53 
54 

55 
56 

57 
58 

i"^ 
(iO 





180 


9 


9 


6 


iS 


o.y 


0. 


7 


21 .0 


I .1 




8 


24.0 


I .2 




9 


27 


1.4 




10 


30.0 


1 6 




20 


00.0 


3-1 




30 


QO.O 


4-7 


4- 


40 


I30.0 


6.3 


6. 


50 


150.0 


7 9 


7 



30 
40 

5" 



8 




8 


0-8 


0.8 1 


I 








9 


I 


I 


I 





I 


3 


I 


2 


I 


4 


I 


3 


2 


§ 


2 


6 


4 


2 


4 





5 


fi 


,s 3 1 


7 


1 


6 


6 1 





7 


6 


6 


0.7 


°f> 1 


7 





8 





7 


8 





9 





8 


9 


I 





1 





10 


I 


I 


1 


1 


20 


2 


3 


2 


I 


3" 


3 


5 


3 


'2 


40 


4 


6 


4 


3 


50 


5 


8 


5 


4 



6 
7 
8 

9 
10 
20 

30 
40 

50 



o 7 
09 



0.6 
0.7 
0.8 
0.9 
1 .0 
2.0 

3-0 
4.0 

5-0 





4 


3 




6 


4 


0-3 


0. 


7 





4 





4 





8 





5 





4 


0. 


9 





6 





5 


0. 


10 





6 





6 


0. 


20 


I 


3 


I 


I 


I . 


3>3 


2 





I 


7 


I 


4° 


2 


6 


2 


3 


2. 


50 


3 


3 


2 


9 


2. 





2 


2 




6 


0.2 


0.2 





7 
8 






3 
3 


0.2 
2 






9 





4 


0.3 


0. 


10 
20 






4 


0-3 
0-6 






30 
40 


I 
I 


2 
6 


I.O 

'•3 


0. 
I. 


50 


2 


I 


1-6 


1 . 



I'. ]' 





?> 


5 


4 


6 


t--5 


0.5 


0.4 


7 


°-6 





6 





5 


8 


7 





$ 





b 


9 


0.8 





7 





7 


10 


0.9 





§ 





7 


20 


!•§ 


I 


6 


I 


5 


30 


2.7 


2 


5 


2 


2 


40 


3-6 


3 


? 


3 





50 


4.t 


4 


I 


3 


7 



397 



TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

8° 9" 



10 

II 

12 
14 



15 
16 

18 
19 



20 

21 

22 

23 
24 



25 
26 

27 
28 
29 



ao 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 
49 



Log. Vers. 



98820 
99000 
99186 
99360 

99539 



•99718 
,99897 
. 0007 5 
,00253 
,00431 



. 00608 
.00784 
. 0096 r 
,01137 
,01313 



T> Lost. Exsec. 



50 

51 

52 
53 
54 



55 

56 
57 
58 
59 



.01488 
.01663 
,01838 
.02012 
.02186 



•02359 
•02533 
,02706 
.02878 
,030^6 



,03222 
•03394 
■03565 
•03736 
03906 



.04076 
,04246 
.04416 

•04585 
•04754 



,04922 
,05090 

■05258 
.05426 

•05593 



m 



,05760 

•05926 
■06093 
.06259 
.06424 



,06589 
,06754 
.06919 
• 07083 
.07247 



.07411 

.07575 
.07738 
.07906 
.08063 



.08225 
,08387 
.08549 
.08710 
,08871 



0Q03 1 



Ijosr. Vers. 



86 
80 

79 
79 

79 
78 
78 
77 
78 

77 
76 
76 
76 
76 

75 
75 
75 
74 
74 

73 
73 
73 
72 
72 
72 
71 
71 
71 
70 

70 

70 

69 
69 

69 

68 
68 
68 
67 
67 
(>7 
66 
66 
66 

65 

65 
65 
65 
64 
64 
64 
63 
63 
62 
62 

62 
61 
62 
61 
61 
66 



7 



99244 
99427 
99609 
99796 
99971 



.00152 
■00332 
,00512 
, 00692 
,00871 



.01050 
.01229 
,01407 
■01585 
,01763 



.01940 
.021 17 
.02293 
,02469 
,0264.!; 



z> 



,02820 
,02995 
,03176 

.03345 
03519 



,03692 
,03866 I 
,04039 1 
,04212 
■ 04384 I 



1) 



«i 
81 
86 
86 
80 
80 
79 

79 
78 
7^ 
78 
77 
77 
77 
76 
76 
75 

75 
75 
75 
74 
74 

73 
73 
73 
73 
72 

72 
71 
71 
71 
70 

70 
70 
70 
69 
69 
69 
68 
68 
68 

67 
67 
67 
66 
66 
66 

65 
65 
65 
64 
64 
64 
63 
64 
63 
63 
62 
8.09569 



•045 56 
,04728 I 
.04899 j 
.05076 I 
.05241 i 



,05411 
,05581 
,05751 
,05921 
. 06090 



,06259 
,06427 
■06595 
.06763 
.06931 



,07098 
.07265 

.07431 
.07598 
.07764 



.07929 
,08095 
,08260 
,08424 
,08589 



.08753 
,08917 
, 0908 I 
,09244 
. 09407 



Loff. Vers. 



8. 0903 T 
.09192 
.09352 
,09512 
,09671 



09836 

09989 

0148 

0306 
0464 



0622 

0779 
0936 
1093 
1250 



1406 
1562 
1718 

1873 
2029 



2184 

2338 
2492 

2647 
2806 



2954 
3107 
3266 
34^3 

3565 



T> Los. Kxsec. 



3717 
3869 
4021 
4172 

4323 



4474 
4625 

4775 
4925 

5075 



5225 
5374 

5523 
5672 
5826 



59^8 
6116 
6264 
6412 
6559 



6706 
6852 

6999 

7145 
729T 



7437 
7582 
7728 

7873 
8017 



8162 



66 
60 

eo 

59 

59 
59 
58 
58 
58 

57 
57 
57 
57 
56 

56 
56 

55 
55 
55 

55 
54 
54 
54 
53 

53 

53 
53 

52 
52 

52 
52 
51 
51 
51 

51 
50 
50 
50 
49 
50 
49 
49 
49 
48 
48 

48 
48 

47 
47 
47 
46 
46 
46 
46 

45 
45 
45 
45 
44 
44 



liOjr. Vers. 



I> 



8 



8 



09569 

09732 

09894 

0056 

0217 



0378 

0539 
0700 
0866 
1026 



1 186 
1340 
1499 
1658 
1816 



1975 

2133 
2291 

2448 
2605 



D 



2762 
2919 

3075 
3232 
338? 



3543 
3698 
3854 
4008 
4163 



4317 
4471 
4625 

4778 
4932 

5085 

523^ 
5390 
5542 
5694 



5846 
5997 
6148 
6299 
6450 



6606 

6750 
6906 
7050 
7199 



7349 
7497 
7646 
7795 
7943 



8091 

8238 
8386 

8533 
8686 



8827 



62 
62 
62 
61 

61 
61 
66 
66 
60 
60 

59 
59 
59 
58 

58 
58 
58 
57 
57 

57 
57 
56 
56 
55 
56 
55 
55 
54 
54 

54 
54 
53 
53 
53 

53 
52 
52 
52 
52 

52 
51 
51 
51 
50 

56 
50 
50 
49 
49 

49 
48 
49 
48 
48 

48 
4? 
47 

a1 

47 
46 



p. P 



10 

1 1 

12 

13 

14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 
30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 



45 

46 

47 

48 

49_ 

50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



(JO 



liOff. Kxseo. /> I 





180 


170 


6 


18.0 


17.0 


7 


21.0 


19-8 


8 


24.0 


22-6 


9 


27.0 


2.S-5 


10 


30.0 


28.3 


20 


60.0 


5^-6 


30 


90.0 


8=i.o 


40 


120.0 


i'3-3 


50 


150.0 


141.6 



20 

30 
40 

50 



30 
40 

50 



40 

5" 



40 
50 



40 
50 



160 

16 .0 
18.6 
21.3 
24.0 
26.6 

53- 

80.0 
ic6.6 
133- 



150 140 

14.0 



21.0 

23-3 
46.6 
70.0 

93-3 
116. 6 



15 





17 


5 


20 





22 


5 


25 





50 





75 





100 


.0 


125 


.0 



0.9 0.9 o 

1.1 1.0 I 

1.2 1.2 1 
1.4 1.3 I 

1 .6 1.5 I 

3-1 3-0 2 

4-7 4-5 4 

6.3 6.0 5 

7-9 7-5 7 



7 

o 7 
0.9 



0.6 



s 


C 


05 





0.6 


0. 


7 


0. 


0.8 





c.g 





1-8 


I . 


2.7 

3-6 
4.6 


3- 
4 



P. p 



398 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

10 11 







6 

7 
8 

9 
10 

II 

12 

13 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



26 

27 



29 



80 

31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 

42 

43 

44 

45 
46 
47 
48 

49 



Lojf. Vers. 1 J> Lop. Kxsec. /> 



8. 18162 
.18306 
.18456 

.18594 
.1S738 



8 



.18881 
. 19024 
.19167 
.19309 

.1945^ 



8.19594 
.19736 
.19878 
.20019 
.20166 



8.20301 
. 20442 
.20582 
.20723 
.20861 



8. 21003 
.21142 
.21282 
.21421 

. 2 [ 560 



8.2169^ 
.21837 
.2197^ 
.221 13 

.22251 



8.22389 
.22526 
.22663 
.22800 
.22937 



8.23073 
.23209 
.23346 
.23481 
.23617 



8.23752 
.23888 
.24023 
.24158 
. 24292 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



8.24425 
.24561 
.24695 
.24828 
. 24962 



25095 
25228 
25361 ' 

25494 
,25627 



(iO 



8.25759 
.25891 
.26023 
.26155 
.26285 



8.2641^ 



Lotf. Vers. I J> 



144 

144 
144 

143 

U3 
143 
142 
142 
142 

142 
142 
142 
141 
141 
141 
146 
146 
146 
140 

140 
139 
139 
139 
139 

138 

'38 

138 
138 

137 
138 
137 
137 
136 
137 

136 
136 

136 

13^ 
136 

135 
135 
135 
135 
134 

134 
134 
134 
133 
133 

133 
133 
133 
132 

133 
132 
132 
132 
132 

131 
131 



8. 18827 
.18973 
.19120 
. 1 9266 
.19411 



8.19557 
.19702 
.19847 
.19992 
.20137 



8.20281 
.20425 
. 20569 
.20713 
.20857 



8.21 000 
.21143 
.21286 
.21428 
.21571 



8.21713 
.21855 

.21995 
.22138 

. 22279 



8.22420 
.22561 
.22701 
.22842 
.22982 



8.23122 
.23262 
.23401 
.23540 

.23679 



8.23818 
.23957 
.24095 
.24234 

.24372 



8.24509 

. 24647 
.24784 
.24922 
.25059 



8.25195 
•25332 

.25468 
.25604 
.25746 



8.25876 
.26012 
.2614^ 
.26282 
.26417 



8.26552 
. 26685 
.26821 
.26955 
.27089 

8. 27223 
iOp. Kxsec. 



146 
146 
146 

145 

145 
'45 
145 
145 
144 
144 
144 
144 
144 
143 

143 
143 
143 
142 
142 

142 
142 
141 
141 
141 

141 
146 
146 
146 
140 

140 
140 

139 
139 
139 
139 
138 
138 
138 
138 

137 
138 

137 
137 
137 

136 
136 
136 
136 
136 

136 

135 

135 

135 

135 

134 

I 134 

' 134 

] 134 

,134 

; 134 
It 



Loff. Vers. 



Jt 



8.26417 
•26548 
.26679 
.26816 
.26941 



8.27071 
.27201 

•27331 
.27461 

.27596 



8.27719 
.27849 

.27977 
.28105 
.28235 



8.28363 
.28491 j 
.28619 
• 28747 
.28S75 



8.29002 
.29129 

■29^56 
.29383 
.29510 



8.29635 
,29763 
.29889 
.30015 
.30146 



8, 



30266 

3039' 
305 1 6 
30642 

30765 



. 3089 1 
.31015 
.31140 
.31264 
.31388 



8.3151T 
.3>635 
.31758 
.31882 

.32005 



8.32128 
.32256 
•32373 
•32495 
.32617 



8.32739 
.32861 

•32983 
•33104 
•33225 



8.33347 
• 33468 

•33588 
. 33709 
•33829 



8.33930 

liOir. Vers. 



131 
131 
131 
136 

130 
130 

130 
130 
129 

129 
129 
128 
I 29 

'28 
128 

128 
128 
128 

127 

'2^ 
127 
127 
127 
126 

126 
126 

126 
126 

125 

125 

125 
125 

125 

124 

124 
124 
124 
124 

124 

123 
124 

'23 
123 
123 
123 

122 

122 
I --) ^ 

122 
122 
122 
I2T 
I2T 
121 

121 
121 
126 
126 
126 
126 

/> 



Kxser. 



n 



8.27223 ^ 

•27356! 
.27490 
.27623 
.27756 



8.27889 
.28021 

.28153 
.28286 
.28418 



8.28550 
.28681 
.28813 
. 28944 
.29075 



8 . 29206 

•29336 
• 29467 
•29597 
. 29727 



8 



.29857 
• 29987 
.30117 
.30245 
.30375 



8 . 30504 

•30633 
.30762 
.30896 
.31019 



8 



• 3"47 

.31275 
.31402 

•31530 
.31657 



8.32418 
•3254-+ 
.32676 

•32796 
.32922 



8.31785' 
.31912 
.32039 
.32165 
.32292 



8.33047 
•33173 
•33298 
•33423 
•33547 



8.33675 
•33797 
•33921 
.34045 
.34169 



8.34293 
.34417 
•34540 
. 34663 
.34785 



33 
33 
33 
33 



j- 



32 



30 



29 

29 

29 

29 
29 

28 

28 
28 

28 
28 

27 
27 

27 

27 
27 

27 

26 
26 

26 

26 
26 
26 

25 

25 
25 
25 
25 
24 

25 
24 

24 

24 

23 

24 

24 







3 

_4 

5 
6 

7 
8 

9 
10 

1 1 
I 2 

13 

i4 

15 
16 

17 
18 

19 



r. 1' 



8 . 3490') 
l.nir. Kxsnr. 



J> 



'JO 

21 



25 
26 

27 
28 

29 

'60 

3' 
32 
33 
1^ 
35 
36 
37 
38 
39 



40 

41 
42 
43 

44 

45 
40 

47 

48 

49 

:>o 

5' 
52 
53 

55 
56 
57 
58 
59 
<>0 



6 


13 


7 
S 


15.1 
J7.5 


9 


19.5 


10 


21.^ 


20 

30 
40 

50 


43-3 

65.0 

86.^ 

108.3 



40 
50 



40 

50 



20 
30 

40 

50 



03 

©•3 
0.4 
0,4 
0.5 
i.o 

2.0 

2-5 



120 

12 .0 
14 .0 
16.0 
18.0 
20.0 
40.0 
60 o 
80.0 
100. o 



3 

0-3 
0.4 
0.4 

0.5 
0.6 
1 .1 



2 
0.2 
0.3 

0.3 
0.4 
0.4 
o.§ 
1.2 






2 


0. 


0.2 


0. 





2 


0. 





3 


0. 





3 


0. 





6 


0. 


I 





0. 


I 


3 


I 


I 


6 


I 



03 
0.4 



I*. V. 



399 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 

13° 13° 



10 

II 

12 

13 
14 



15 
i6 

i8 
19 



20 

21 

22 
23 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 
60 



Los. Vers. 



33950 
34070 

34190 
34309 
34429 



34549 
34668 

34787 
34906 

35025 



35143 
35262 

35386 

35498 

35616 



35734 
35852 

35969 
36086 
36204 



36321 
3643? 
36554 
36671 

36787 



36903 
37019 
37135 
37251 
37366 



37482 

3759^ 
37712 
3782^ 
37942 



38057 

38171 
382S6 
38400 
38514 



38628 
38741 
38855 
38969 
39082 



39195 
39308 
39421 
39534 
39646 



39758 
39871 

39983 
40095 
40207 



D 



40318 
40430 
40541 
40652 

40764 
8.40875 
Lour. Vers, i I) 



20 
20 

19 

20 

9 
9 
9 
9 
9 



Lojf. Exsec. 2> 



8 



34909 
35032 

35155 
35277 
35399 



35522 
35644 
35765 
35887 
36009 



36130 
36251 
36372 

36493 
36614 



36734 
36855 
36975 
37095 

37215 



37931 
38050 
38169 
38287 
38406 



38524 
38642 
38760 
38878 
38995 



39113 
39230 
39347 
39464 
3958T 



39698 
39814 
39931 
4004^ 
40163 



40279 

40395 
405 II 

40626 

40742 



40857 
40972 
4108^ 
41202 
41317 



37335 
37454 
37574 
37693 
37812 



41431 
41546 
41666 

41774 
41888 
42002 

jOfj. Kxsec. 



23 

22 
22 
22 

22 
22 
21 
22 
21 

21 
21 

21 
26 
21 

26 
26 
20 
20 
20 
20 

9 
9 

9 
9 
9 
8 
9 
8 
8 



8 
J 

1 
7 
7 
7 
7 

6 
6 
6 
6 
6 

6 
6 

5 
5 
5 

5 
5 
5 
5 
4 

4 
4 
4 
4 
4 
4 

IF 



Log. Vers. | It |Loif. Exsec. 



40875 
40985 
41096 

412O6 
41317 



41427 

4153^ 
4164^ 

41757 
41867 



41976 
42086 
42195 

42304 

42413 



42522 
42636 

42739 
4284^ 
42956 



43064 
43172 
43280 
43388 
43495 



43603 
43710 

43817 
43924 
44031 



44138 

44245 

44351 
44458 

44564 



44670 

44776 
44882 

44988 
45093 



45 '99 
45304 
45409 
45514 
45619 



45724 
45829 

45934 
46038 
46142 



46247 
46351 
46455 
46558 
46662 



46766 
46869 
46972 
47076 
47179 
47282 

Lotf. Vers. 



10 
16 
16 
16 
16 
10 
10 

09 
10 

09 
09 
09 
09 
09 
09 

08 
09 
08 
08 
08 
08 
08 
08 
07 

07 
07 

07 
07 
07 

06 
07 
06 

06 
06 

06 

06 

05 
06 

05 

05 
05 
05 

05 
05 

05 

04 

05 
04 

04 
04 
04 
04 
03 
04 

03 
03 
03 
03 
03 
03 

"77" 



42002 
42 II 6 
42229 
42343 
42456 



42569 
42682 

42795 
42908 
43021 



43133 
43246 
43358 
43470 
43582 



43694 
43805 

43917 
44028 
44'39 



44251 
44362 
44473 
44583 
44694 



44804 

44915 
45025 

45135 
45245 



45355 
45465 
45574 
45684 

45793 



45902 
4601 T 
46126 
46229 
46338 



46446 

46555 
46663 

46771 
46879 



46987 

47095 
47203 

47316 
47417 



47525 
47632 

47739 
47846 
47953 



48060 
48166 

48273 
48379 
48485 
8.48591 

Losj. Exsec. 



D 



13 

13 
13 
13 

13 
13 

'3 
13 
12 

12 
12 
12 
12 
12 

12 

i" 

I 

I 

I 

I 

I 

I 

16 

16 

16 
16 
10 
16 

09 
10 
10 
09 
09 
09 
09 
09 
09 
08 
09 
08 
08 
08 

08 
08 

08 

07 
08 
07 
07 
07 
07 
07 
07 
06 
07 
05 
06 
06 

06 

06 



10 

II 

12 

13 

14 



15 
16 

17 
18 

19 



20 

21 

22 

23 

24 



25 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 

46 

47 
48 

49 



50 

51 
52 
53 

55 
56 
57 
58 
59 
(50 



P. P 





120 


119 


6 


12,0 


II. 9 


7 
8 

9 


14.0 
16.0 
18.0 


139 
15.8 
17-8 


10 


20.0 


19-8 


20 
30 
40 


40.0 
60.0 
80.0 


39-6 
59-5 
79-3 


50 


100. 


99.1 



117 

II. 7 

13-6 
15.6 

175 
19-5 
39 o 
58.5 
78.0 

97.5 



13-5 
15-4 
17.4 

"9-3 
38.6 
58 o 

77-3 
96.6 



118 



116 115 



II-5 
13.4 

^5-3 
17.2 
iQ.i 
38.3 

57-5 
76.6 
95.8 





II 


4 


113 


11: 


6 


11.4 


"•3 


II. 


7 


13 


3 


13.2 


13- 


8 


TS 


2 


15.0 


14- 


9 


17 


I 


16.3 


16. 


10 


19 





18.8 


18. 


20 


38 





37-6 


37 


30 


57 





5'j-5 


5b. 


40 


76 





75.3 


74- 


50 


95 





94.1 


93- 





III 


IIO 


6 


11 .1 


11. 


7 


12.9 


12.8 


8 


14.8 


14-6 


9 


16.6 


16.5 


10 


18.5 


18.3 


20 


37-0 


36-6 


30 


55-5 


55-0 


40 


74.0 


73-3 


SO 


92-5 


91-6 



109 



14-5 
16.3 
18.1 
36.3 
54-5 
72-6 
90.8 



6 


108 

10.8 


107 

10.7 


7 


12.6 


12-5 


8 
9 


14.4 
16.2 


14.2 
16.0 


10 

20 


18.0 
36.0 


17-8 
35-6 


30 


54 


53-5 


40 

50 


72.0 
90.0 


89.1 



106 

10.6 





105 


104 


6 


10.5 


10.4 1 


7 


12.2 


12 


I 


8 


14.0 


13 


8 


9 


'5-7 


13 


fi 


10 


17-5 


17 


3 


20 


35 -o 


34 


6 


33 


52.5 


52 





40 


70.0 


tq 


3 


50 


87.5 


86 


6 



0.0 
0.0 
0.0 

O.I 
O.I 
O. I 

0.2 

0-3 
0.4 



P. l* 



400 



TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

14° 15" 







10 

1 1 

12 

J3 
U 



15 
i6 

i8 
19 



20 

21 

22 

23 

24 



25 
26 

27 
28 
29 



80 

31 
32 

34 



35 
36 

37 
38 
39 



40 

41 
42 
43 

44 



45 
46 

47 
48 
49 
50 

51 
52 
53 
54 



55 

56 
57 
58 
59 



00 



Loir. V«>rs. 



I) 



8.47282 
47384 
47487 
47590 
47692 



8 



47795 
47897 

47999 
48 1 01 

48203 



48304 
48406 

48507 
48609 
48716 



4881 1 
48912 

49013 
491 14 
49215 



49315 
494 Id 
49516 
49616 
49716 



49816 
49916 
50015 
501 1 5 
50215 



50314 

50413 
50512 

5061 1 
50710 



50809 
50908 
51006 
51105 
51203 



5 1 301 

51399 
51497 
51595 
51693 



51791 
51888 
519.S6 
52083 
52180 



52277 
52374 
52471 
52568 
52665 



52761 
52858 

52954 
53050 
53146 



53242 



02 

03 
02 
02 

02 
02 
02 
02 
02 

01 
01 
01 
01 
01 

01 
01 
01 

CO 

01 

06 
00 
00 
00 
00 

00 
00 

99 
100 

99 
99 
99 
99 
99 
99 

98 
99 
98 
98 
98 
98 
98 
98 
98 
97 
98 
97 
97 
97 
97 
97 
97 
97 
96 
97 

96 

96 

96 

96 
96 
96 



Loir. KxN«M'. 



/> 



Loir. Vers. 



/> 



8 



48591 
48697 
48803 
48909 
49014 



49120 
49225 

49331 
49436 
4954 » 



49646 
49750 

49855 
49960 
50064 



50168 
50273 
50377 
50481 

50585 



50688 

50792 
50896 

50999 
51 102 



51205 

51309 
51412 

51514 
51617 



51720 
51822 
51925 
52027 
52129 



52231 
52333 
52435 
52537 
52638 



52740 
5284T 
52943 
53044 
53'45 



53246 
53347 
53448 
53548 
53649 



53749 
53850 
53950 
54050 
54150 



54250 

54350 

54449 

54549 
54649 



54748 

Kxser. 



06 
06 
05 

05 

05 
05 
05 
05 
05 

05 
04 

05 
04 

04 

04 

04 

04 

04 

04 

03 
04 
03 
03 
03 

03 

03 

03 
02 

03 
02 
02 
02 
02 
02 

02 
02 
02 

oT 
01 

01 
01 
of 
01 
01 

01 
01 
01 
00 
06 

06 
06 
00 
00 
00 

00 
00 

99 
100 

99 
99 



/> 



Loir. Vers. 



8 



8 



53242 

5^ ^ ^ o 
jjj8 

53434 
53530 
53625 
53721 
53816 
53911 
54007 
54102 



/> 



54197 
54291 
54386 
54481 
54575 



54670 
54764 
54858 
54952 
55046 



55140 

55234 
55328 
55421 
55515 



55608 
55701 

55795 
55888 
55981 



56074 
56166 

56259 

56352 
56444 



56536 
56629 

56721 

56813 

56905 



56997 
57089 
57186 
57272 
57363 



57455 
57546 
57637 
57728 
57819 

57910 
58001 
58092 
58182 

58273 
58363 
58453 
58544 
58634 
58724 



;88i4 



96 
95 
96 
95 

95 
95 
95 
95 
95 

95 
94 

95 
94 
94 

94 
94 
94 
94 
94 
94 
93 
94 
93 
93 

93 
93 
93 
93 
93 

93 
92 
92 

93 
92 

92 

92 

92 

92 

92 

92 
92 
91 
91 
91 

91 
91 
91 
91 
91 

91 
96 

91 
90 
96 

90 
90 
96 
90 
90 
90 



Lojr. Vers. 



/> 




/> 



8 
Lmr 



55736 
55834 

55933 
56031 
56129 



56226 

56324 
56422 
56519 
56617 



56714 
56812 

56909 
57006 

57103 



57200 
57296 
57393 
57490 
57586 



57682 

57779 
57875 
57971 
c;8o67 



58163 
58259 

58354 
58450 
58546 



58641 

58736 
58832 

58927 
59022 



59117 
592 1 T 

59306 
59401 

5949^ 



59590 
59684 

59779 
59873 
59967 

60061 
60)55 
60249 
60342 
60436 
60 5 30 



99 
99 
99 
99 

99 
99 

98 
98 
98 

98 
98 

98 
98 

98 

97 
98 

97 
97 
97 
97 
97 
97 
97 
97 
97 
96 
97 
96 
96 
96 

96 
96 

96 
96 

95 
90 

95 
96 
95 

95 
95 
95 
95 
95 

95 

94 
95 
94 
94 
94 
94 
94 
94 
94 

94 
94 

94 
93 
94 
93 



10 

1 1 
I 2 

13 
14 

15 

16 

17 
18 

19 



20 

21 



24 



25 
26 

27 
28 

29 

30 

31 

32 
33 
34 

35 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 

i9 
50 

51 
52 
53 

54 



55 
56 

57 
58 



V. V 



103 

10.3 



68.6 



102 

10 



lOI 



II 


9 


11.8 


13 


6 


13.4 


15 


3 


15.: 


17 





.6.fi 


34 





33-6 


5» 





50-5 


68 





67. H 


85 





84.1 





100 


99 


6 


1 0.0 


9.9 


7 


II. 6 


"•5 


8 


'3-3 


13.2 


9 
10 


15-0 
16.6 


14.8 
16.5 


20 


33-3 


33-0 


30 
40 


50.0 
66.6 


4Q-5 
66.0 


50 


83.3 


82.5 



98 

9.S 

1 1. 4 

13.0 
14.7 
,6.5 

3«.6 
49.0 

65.3 
8:. 6 



6 


9 

. 9 


7 

7 


96 

9.6 


95 

9-5 


7 


II 


3 


II. 2 


11. 1 


8 


12 


9 


12.8 


12.6 


9 
10 


14 
16 


i 


14.4 
16.0 


14.2 

»5-8 


20 
30 
40 


32 
48 
64 


3 
5 


32.0 
48.0 
64.0 


3«-6 
47 -5 
633 


50 


80 


8 


80.0 


79.1 





94 


93 


b 


9.4 


9-i 1 


7 


10.9 


10.8 


8 


12.5 


12.4 


9 


14. 1 


«3-9 


10 


>5-0 


15-5 


20 


31-3 


31.0 


30 


47.0 


46.5 


40 


62.6 


62.0 


50 


78.3 


77.5 



92 

Q.2 
10.7 
12.2 
13.8 

J5 3 
30-6 
46.0 
61.3 

76.6 





91 


90 


6 


9-1 


9.0 


7 


10.6 


10.5 


8 


12. I 


13.0 


9 


'3-6 


»3o 


10 


J5-I 


'5 


20 


30-3 


30.0 


^0 


45-5 


45.0 


40 


60.^ 


60.0 


50 


75-8 


75.0 



0.11 
06 
0.0 

O. I 
O.I 

o.i 

o.a 

o 3 

0.4 



^!^s('(• 



/> I 



401 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



16 



11° 



10 

II 

12 

13 

14 



15 

i6 

17 
i8 

19 



20 

21 
22 

23 
24 



25 
26 
27 



29 



80 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



Loa:. Vers. 



50 

51 

52 
53 
54 



55 
56 

57 
58 
59 

(io 



8.58814 
58904 

58993 
59083 
59173 



z> 



8 



59262 

59351 
59441 

59530 
59619 



59708 

59797 
59886 

59974 
60063 



60152 
60246 
60328 
60417 
60505 



60593 
60681 
60769 
60857 
60944 



61032 
61119 
61207 
61294 
6138T 



61469 
61556 
61643 
61730 
61816 



61903 
61990 
62076 
62163 
62249 



62336 
62422 
62508 
62594 
62680 



62766 
62852 
62937 
63023 
63108 



63194 
63279 

63364 
63449 
63534 



63619 
63704 
63789 
63874 
63959 
8 . 64043 

Lost. Veix. 7> 



90 
89 
90 



89 
89 

89 
89 

89 
89 
89 

88 
89 



8S 
88 



87 

Sf 
87 
87 

87 
87 
87 
87 
86 
87 
86 
86 
86 



86 
86 
86 
86 

86 
86 



85 
85 
85 
85 
85 

85 
85 
85 
84 
85 
84 



Loff. Kxsec 



8.60530 
.60623 
.607 16 
.60810 
. 60903 



8.60996 
.61089 
.61182 
.61275 
.61368 



8.61466 

•61553 
.61645 

.61738 
.61830 



8.61922 
.62014 
.621O6 
.62198 
.62296 



8.62382 
.62474 
.62565 
.62657 
.62748 



n 



i. 62840 
.62931 
.63022 
.63113 
.63204 



8.63295 
.63386 
.63477 
.63567 
.63658 



8.63748 

.63839 
.63929 
.64019 
.64109 



8.64199 
.64289 

•64379 
. 64469 

.64559 



8 . 64649 

•64738 
.64828 
.6491^ 
.65006 



8.65096 
.65185 
.65274 

•65363 
.65452 



8.65541 
.65629 
.65718 
.65807 
•65895 

8.65984 



93 
93 
93 
93 

93 
93 
93 
92 
93 

92 
92 

92 
92 
92 

92 
92 
92 

92 
92 

91 
92 
91 
91 
91 

9^ 
91 
91 
91 
91 
96 

91 

91 
96 

90 

90 
96 
90 
90 
90 
90 
90 
90 
90 
89 

90 
89 
89 
89 
89 

89 
89 
89 
89 
89 
89 
88 
88 
89 

88 
88 



Kxsec. I J) 



Log 
8. 



Vers. 



64043 
64128 
64212 
64296 
64381 



,64465 
,64549 

64633 
,64717 

,64801 



,64884 
, 64968 
,65052 

.65135 
.652I8 



,65302 
.65385 

.65468 
,65551 

•65634 



,65717 
,65806 
,65883 
,65965 
, 66048 



.66131 
.66213 
.66295 
.66378 
, 66460 



.66542 
.66624 
,66706 
,66788 
,66870 



,66951 
67033 
,67115 

•67196 

,67277 



67359 
.67446 
.67521 
.67602 
•67683 



,67764 

,67845 
67926 
, 68007 
,6808^ 



,68168 

.68248 
.68329 
. 68409 
,68489 



8.68569 
.68650 
.68730 
.68810 
.68889 

8.68969 

Loir. AVrs. 



n 



84 
84 
84 
84 

84 
84 
84 
84 
84 

83 
83 
84 

83 
83 

83 
83 
83 
83 

83 

83 
83 
82 

82 

83 

82 



82 



81 
81 
82 
81 
81 

81 
81 
81 
81 
81 

81 
81 
86 
81 
86 

86 

86 

86 

86 

80 

80 

86 

80 

80 

79 
80 

"tT" 



Lost. Kxsec. 



8. 



65984 
66072 
66166 
66248 
66336 



,66425 
,66512 
. 66606 
.66688 
.66776 



8. 



66863 
66951 
67039 
67126 
67213 



8. 



01 



67j 
67388 

67475 
67562 

67649 



,67736 
,67822 
,67909 
.67996 
,68082 



,68169 
,68255 
,68341 
,68428 
.68514 



. 68600 
.68686 
.68772 
,68858 
, 68944 



.69029 
.69115 
.69201 
,69286 
.69372 



69457 
69542 
,6962^ 
,69712 
,69798 



69883 

6996^ 

70052 

,7013^ 

,70222 



70306 
70391 

70475 
,70560 
70644 



8.70728 
.70813 
. 70897 
.70981 
.71065 

8.71149 

iOjr. Kxspc. 



i» 



88 
88 

88 
8? 
88 
88 
87 

8? 
88 

S? 
87 
8? 
8f 
87 
87 
87 
87 
87 
86 
87 
86 
86 

86 
86 
86 

86 
86 

86 

86 

85 
86 

86 

85 
86 

85 

85 
85 

85 

85 

85 
85 

85 

85 
84 

85 
85 
84 

84 
84 
84 
84 
84 
84 
84 
84 
84 
84 
84 

"77" 



p. P. 



10 

II 
12 

13 
14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 
27 
28 
29 



30 

31 
32 

34 



35 
36 
37 
38 
39 



40 

41 
42 

43 
44 



45 
46 
47 
48 
49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 
CO 





93 


92 


6 


9-3 


9.2 


7 


10. 8 


10.7 


8 


12.4 


t2.2 


9 


13-9 


13-8 


10 


15-5 


15.3 


20 


31.0 


30.6 


^0 


46.5 


46.0 


40 


62.0 


61.3 


50 


n-s 


76.6 





90 


89 


6 


9.0 


8.9 


7 


10.5 


10.4 


8 


12.0 


II. 8 


9 


13^5 


13-3 


10 


15.0 


14-8 


20 


30.0, 


29^6 


30 


450 


44-5 


40 


60.0 


59-3 


50 


75-0 


74.1 





87 


86 


6 


8.7 


8.6 


7 


10. 1 


10. 


8 


II. 6 


II. 4 


9 


13.0 


12.9 


10 


14-5 


14.3 


20 


29.0 


28.6 


30 


43-5 


43^o 


40 


58.0 


57-3 


50 


T^'l 


71-6 





84 


83 


6 


8.4 


8.3 


7 


9.8 


9-7 


8 


II. 2 


1 1.O 


9 


12.6 


12.4' 


10 


14.0 


13.8 


20 


28.0 


27.6 


30 


42.0 


41.5 


40 


56.0 


55.3 


50 


70.0 


69.1 





81 


80 


6 


8.1 


8.0 


7 
8 


9.4 
10.8 


9-3 
10.6 


9 


12.1 


12.0 


10 
20 


13-5 
27 


133 
26.6 


3<3 

40 
50 


40-5 
54-0 
67.5 


40.0 

53-3 
66.6 



7 
8 


0.0 
0.0 


9 
10 


I 
0.1 


20 


0.1 


30 
40 

50 


0.2 

0-3 
0.4 



91 

9- 
10. 1 
12. 
13-1 
15- 
Z°' 
45- 
60. 



75-8 



88 



II. 7 
13.2 

14-6 

29-3 
44.0 

58.6 
73- 



85 

8.S 
9.9 

"•I 
12.7 

14. 

28.3 
42.5 
56.6 
70.8 



82 

8.2 

9-5 
10.9 
12.3 

13-6 

27. 

41. 

54-6 
68.3 



79 

7-9 

9- 
10.5 
II . 

26.3 
39-5 
52 6 
65^8 



P. P 



402 



Table viil— logarithmic versed sines and external secants. 



18' 



19 



10 

1 1 

12 

13 
14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 

24 



25 
26 

27 

28 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



10 

41 

42 

43 
44 



45 
46 
47 
48 

49 



50 

51 
52 
53 
54 



55 
56 

57 
58 
59 



Loff. Vers, i 2> 



8.68969 
.69049 
.69129 
.6920^ 
.69288 



8.69367 

.69446 
.69526 
,69605 
.69684 



8.69763 
.69842 
.69921 
. 70000 
. 70079 



8.70157 
.70236 
.70314 
.70393 
.70471 



8,70550 
.70628 
.707O6 
.70784 
. 70862 



8 . 70946 

.71018 
.71096 

.71174 
.71251 



71329 
71406 
,71484 
,71561 
,71639 



71716 

.71793 
.71876 

■71947 
.72024 



8.7210T 
.72178 
.72255 
.72331 
.72408 



8.72485 
.72561 
,72637 
.72714 
,72796 



GO 



8.72866 
.72942 

.73018 
•73094 
,73176 



8.73246 
.73322 

•73398 
•73473 
•73549 



8.73625 



Log. Vers 



79 
80 

79 
79 

79 
79 
79 
79 
79 
79 
79 
79 
7^ 
79 

78 
78 
78 
78 
78 

7^ 
78 

78 
78 
78 

78 
78 
77 
78 
77 
71 
7l 
77 
77 
77 

77 
77 
77 
77 
77 

77 
7l 
77 
76 
77 

7l 
76 

76 

76 
76 
76 
76 
76 
76 
76 

76 
76 

7% 
71 
76 
7% 



Log. Kxsec. 



.71149 
.71232 

.713I6 

, 7 1 400 
,71484 



J* 



8.71567 
,71651 

.71734 
,71817 
. 7 1 90 1 



8,71984 
.72067 
.72150 
.72233 
.72316 



8.72399 
.72481 
.72564 
.72647 
.72729 



8.72812 
.72894 

.72977 

•73059 
.73141 



8.73223 
.73306 
.73388 
.73470 
.73551 



8.73633 
•73715 
.73797 

.73878 
.73960 



74041 
.74123 
. 74204 
,74286 
.74367 



8.74448 

.74529 
.74616 

.74691 

.74772 



8.74853 
.74934 
.75014 
.75095 

.75175 



8.75256 

•75336 
•75417 
.7549? 
•75577 



8.75658 
•75738 
.75818 

.75898 
•75978 



8.76058 



J> IliOSf. Kxsec 



83 
84 
83 
84 

83 
83 
83 
83 
83 

83 
83 
83 
83 
83 

83 
82 

83 
82 
82 

82 
82 
82 
82 
82 

82 
82 



81 

82 
82 
81 
81 
81 

81 
81 
81 
81 
81 

81 
81 
81 
81 

86 

81 
81 
86 
86 
86 

86 
86 
86 
86 
80 
86 
80 
80 
80 
80 
80 



Log. \ 



/> 



8 



7> 



73^23 
73700 

73775 
73851 
73926 



74001 
74076 
74151 
74226 

7430T 



74376 

74451 
74526 

74606 
74675 



74749 
74824 

74898 
74973 
75047 



75121 

75195 
75269 

75343 
75417 



75491 
75565 
75639 
75712 

75786 



75860 

75933 
76006 
76080 

76153 



76226 
76300 

76373 
76446 

76519 



76592 
76664 

7673? 
76810 
76883 



76955 
77028 
77106 
77^72> 
772AS 



771>^7 
77390 
77462 

77534 
77606 



77678 
77750 
77822 

77893 
77965 



78037 



75 
75 
75 
75 

75 
75 
75 
75 
75 

75 
74 
75 
74 
7-1 

74 
71 
74 
7l 
74 

74 
74 
74 
74 
74 

74 
73 
73 
73 
73 
74 
73 
74 
73 
73 

73 
73 
73 
73 
73 

73 
72 
73 
72 

73 

72 
72 
72 
72 
72 

72 
72 
72 
72 
72 
72 
72 
72 
71 
72 
71 



Log. KxKec. 



Log. Vers. -?> 



76058 
76137 
76217 
76297 
76376 



76456 
76536 
76615 
76694 
76774 



76853 

76932 
7701T 
77096 
77169 



48 

.2? 



77 

77?>'^ 

77406 

77485 
77563 



77642 

77799 
77^77 
77956 



It 



78034 
78112 
78191 
78269 
78347 



78425 
78503 
78581 

78659 
7873S 



78814 
78892 

78969 
79047 
79124 



79202 

79279 
79357 
79434 
79511 



79588 
79665 
79742 
79819 

79896 



79973 
80050 

80126 
80203 
80280 



80356 
80433 
80509 
80586 
S0662 



8.80738 



79 
80 

79 
79 
80 

79 
79 
79 
79 
79 
79 
79 
79 
79 

79 
79 
78 
79 
78 
78 
78 
79 
78 
78 

78 
78 
78 
78 
78 

78 
78 
78 
78 
71 
78 
7l 
7l 
71 
7l 

7l 
7l 
7l 
77 
77 

7l 
77 
77 
77 
77 

76 
77 
76 
77 
76 
76 
76 
76 
76 
76 
76 



liOir. Kxser.l i> 



15 
16 

17 
18 

19 



20 

21 

22 

23 

24 



25 
26 

27 
28 

29 



30 



32 



34 



35 
36 
37 
38 
39 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



P. 1*. 





84 


83 


6 


84 


8.j 


7 


9 


8 


9-7 


8 


II 


2 


11. 


9 


12 


6 


12.4 


10 


>4 





13-8 


20 


28 





27 6 


30 


42 





4^ 5 


40 


56 





55-3 


50 


70 





69.1 





81 


80 


6 


8.1 


8.0 


7 
8 


9.4 
10.8 


9^3 
10.6 


9 


12 I 


12.0 


10 


'3.5 


i3^3 


20 


27.0 


26.6 


30 


40.5 


40.0 


40 
50 


54.0 
67.5 


53 • 3 
66.6 





78 


77 


6 


7.8 


7.7 


7 


9.1 


9.0 


8 


10.4 


10.2 


9 


II. 7 


"•5 


10 


13.0 


12. § 


20 


26.0 


25-6 


30 


39.0 


38.5 


40 


52 .0 


5'-3 


50 


65.0 


64.1 





75 


74 


6 


7.5 


7^4 


7 


8.7 


8.6 


8 


10.0 


9-8 


9 


II. 2 


II .1 


10 


12.5 


12 3 


20 


25.0 


24-6 


30 


37.5 


37.0 


40 


50.0 


49.3 


50 


62.5 


61.6 



72 71 



7.2 


7.« 


8.4 


8.3 


9.6 


9.4 


10.8 


JO. 6 


12.0 


"8 


24.0 


23^ 


36.0 


3S^5 


48.0 


*7-3 


60.0 


59.1 



82 

8.2 j 

9-5 : 

10. y ' 
12.3 ' 
»3.6 
27^3 
41 .0 

54.6 
68.3 



79 

7.9 
9.2 

10 5 

!'.§ 
13.1 
26.3 

39-5 
52.6 
65-3 



76 

7.6 



II. 4 

ia.§ 

25.3 
38.0 

63.3 



73 

7^3 
8.5 

9-7 
10.9 

12. 

24. 

36. 

48.6 

60.8 



0.3 
0.4 



r. i'. 



403 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

20° 21° 







Log. Vers. 



10 

II 

12 

13 
14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 
27 



29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



8.78037 
.78108 
.78180 
.78251 
.78323 



8.78394 
.78466 

.78537 
. 78608 
.78679 



8.78753 
.78821 
.78892 
.78963 
•79034 



8, 



79105 

79175 
79246 
79317 
79387 



jy Log. Exsec. X> 



79458 
79528 

79598 
79669 

79739 



79809 
,79879 

•79949 
.80019 
. 80089 



80159 
.80229 
. 80299 
• 80369 
■ 80438 



80508 
8057^ 
, 80647 

.807 16 
. 80786 



.80855 
. 80924 
.80993 
.81063 
.81132 



(>0 



81201 
.81270 

■81339 
. 8 1 407 

•81476 



81545 
.81614 
,81682 

.81751 
,81819 



8.81888 

•81956 
.82025 
.82093 
.82161 
8.82229 
Log. Vers. 



71 
71 
71 
71 
71 
71 
71 
71 
71 

71 
71 
71 

71 

70 

71 
70 

71 
70 
70 

70 

70 
70 
70 
70 
70 
70 
70 
70 
70 

70 
70 
69 
70 
69 

69 
69 
69 
69 
69 

69 
69 
69 
69 
69 

69 
69 
69 

68 
69 

68 
69 
68 
68 
68 

68 
68 
68 
68 

68 
68 



8 



.80738 
.80814 
.80891 
. 80967 
•81043 



8.81119 
.81195 
.81271 

.81346 
.81422 



8.81498 

.81573 
.81649 
.81725 
.81806 



8.81876 
.81951 
.82025 
.82102 
.82177 



82252 
82327 
82402 

8247^ 
82552 



.82627 
.82702 
.82776 
.82851 
.82926 



. 83006 

.83075 
.83149 
•83224 

.83298 



•83373 
•83447 
.83521 

.83595 
.83670 



.83744 
.83818 
.83892 
.83966 
• 84039 



8.84113 
.84187 
.84261 

.84334 
. 84408 



.84481 

.84555 
.84628 
. 84702 

•84775 



8.84848 
.84922 

.84995 

.85068 

.8514^ 

8.85214 

Log. Kxsec. 



76 

76 
76 
76 
76 
76 
76 

75 
76 

75 

75 

76 

75 
75 

75 
75 
75 

75 
75 

75 
75 
75 
75 
74 

75 
75 
74 
75 
74 

74 
74 
74 
74 
74 

74 
74 
74 
74 
74 
74 
74 
74 
74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
73 
73 

73 
73 
73 
73 
73 
73 



Log. Vers. 



82229 
.8229^ 
,82366 
.82434 
.82502 



82569 
8263^ 
82705 
82773 
82841 



829O8 
,82976 

. 83043 
,83111 

83178 



83246 

83313 
83386 

8344^ 
83515 



,83582 
,83649 
,83716 

.83783 
83850 



839I6 
.83983 
.84050 
.84117 
.84183 



D Loe. Exsec. 



84250 
843I6 
.84383 
. 84449 
.84515 



84582 

84648 
,84714 
, 84786 
,84846 



,84912 

84978 
.85044 
.85116 
.85176 



85242 
85308 
85373 
85439 
85505 



85570 
85626 
85701 

85766 
85832 



8.85897 
.85962 
.8602^ 
. 86092 
.86158 

8.86223 
J) \ Lost. Vers. 



68 

68 
68 
68 

6? 
68 
68 

6? 
68 

67 
6J 

6? 
6^ 
6? 
6J 

67 
6J 

67 
6J 

67 
67 

67 

67 
67 

66 
67 
66 
67 
66 

66 
66 
66 
66 
66 

66 
66 
66 
66 
66 

66 
66 
66 
66 
66 

65 
66 

65 

65 
66 

65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 



8. 



85214 
8528^ 
85366 

85433 
85506 



85579 
85651 
85724 

85797 
85869 



,85942 
86014 
86087 
,86159 
,86231 



86304 
86376 
86448 
86526 
86592 



86664 

86736 
86808 
,86886 
86952 



87024 

87095 
,8716^ 

.87239 
,87316 



87382 
87453 
87525 
87596 
87668 



87739 
,87816 

.87881 

■87953 
. 88024 



z> 



88095 
88166 
88237 
,88308 
.88378 



88449 
88526 
,88591 
,88661 
,88732 



88803 
,88873 

88944 
89014 
,89085 



8.89155 
.89225 

.89295 
.89366 

.89436 

8.89506 

t) Log. Exsec. 



73 
73 

72 

73 

73 
72 

73 
72 
72 

72 
72 
72 
72 
72 
72 

72 
72 
72 
72 

72 
72 
72 
72 

71 
72 
71 
72 
71 
71 
71 
71 
71 
71 
71 

71 
71 
71 
71 
71 
71 
71 
71 
71 
76 

71 
71 
76 

70 
71 
76 
76 

70 
76 
76 

70 
76 

70 
76 

70 
70 

n 



10 

II 
12 

13 

14 

15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 
30 

31 
32 
33 

34 



35 
36 

37 
38 
39 



r. p. 



50 

51 

52 
53 

55 
56 
57 
58 

i9^ 
60 



76 75 74 



6 


7.6 


7.5 


7 


«-8 


8.7 


« 


10. 1 


10.0 


9 


11.4 


11.2 


10 


12 6 


12.5 


20 


25-3 


25.0 


30 


38.0 


37.5 


40 


50.6 


50.0 


50 


63.3 


62.5 



20 

30 
40 

50 



7.4 



12.3 

24.6 

37-0 
49-3 
61.5 





73 


72 


6 

7 
8 


7.3 
8.5 
9.7 


7.2 
8.4 
9.6 


9 


10.9 


10.8 


10 


12.1 


12.0 


20 
30 
40 
50 


24-3 

36.5 
48.6 
60. § 


24.0 
36.0 
48.0 
60.0 



6 

7 


70 

7.0 
8.1 


69 

6.9 
8.5 


8 


9.3 


9.2 


9 


10.5 


10.3 


10 


II. 6 


II. 5 


20 


233 


23.0 


30 
40 

50 


35-0 
4C.6 
58.3 


34.5 
46.0 

57-5 



71 

7.1 

8.3 

9.4 

10. 6 
II. § 
23.6 
35-5 
47.3 
59- 



68 

6.8 

7-9 
9.0 

IO-2 

".§ 
22.6 

34-0 
45-3 
56-6 



6 


67 

6.7 


66 

6.6 


6« 

6. 


7 
8 


7.8 
8.9 


7.7 
8.8 


7- 
8. 


9 


10. 


9.9 


9. 


10 


II .1 


II. 


10 


20 


22.3 


22.0 


21. 


30 


33-5 


33.0 


32 


40 


44.6 


44.0 


43 • 


50 


55.8 


55.0 


54- 



P. p 



404 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

22° 2:r 







10 

1 1 

12 



15 
i6 

17 

i8 
19 



20 

21 
22 

23 

24 



-5 

26 

27 
28 
29 



30 

31 
32 

34 



J3 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
4S 
49 



50 

51 
52 
53 

54 



Loj?. Vers. ! J* 



8.S6223 
.86287 
.86352 

.86417 
.86482 



8.86547 
.86612 

.86676 
.86741 
.86805 



8.86870 
.86934 
. 86999 
.87063; 

.87127 I 



8.87192 
.87256 
.87326 
.87384 
• 87448 



8.87512 

•87576 
.87640 

.87704 

.87768 



8.87832 
.87895 

.87959 
.88023 

. 88085 



8.88150 
.88213 
.88277 
.88340 
. 88404 



8.88467 
.88536 
.88593 
.88656 
.88720 



8.88783 
.88846 
.88909 
.88971 
• 89034 



8 . 89097 
.89160 
•89223 
.89285 
•89348 

8.8941 1 

•89473 
.89536 

.89598 
. 89666 



55 
56 
57 
58 
59 
(>0 



8.89723 
•89785 

•89847 
.89910 

•89972 
8 • 90Q34 

IjOj;. Vers. I 7> 



64 
65 
65 
65 
64 
65 
64 
64 
64 

64 
64 
64 
64 
64 
64 
64 
64 
64 
64 

64 
64 
64 
64 
63 
64 
63 
64 
63 
63 

63 
63 
63 
63 
63 

65 
63 
63 
63 
63 

63 

63 

63 
62 

63 

63 
62 

63 
62 
62 

63 
62 
62 
62 
62 

62 
62 
62 
62 
62 
62 



Kxsec I> 



8 



89506 

89576 
89646 

897 16 
89786 



89856 
89926 
89995 
90065 
90135 



90205 I 
90274 I 
90344 
90413 
90483 I 



90552 
90622 
90691 
90766 
90830 



90899 
90968 
91037 
91106 
91175 



91244 

91313 
91382 
91451 
91520 



91588 
91657 
91726 

91794 
91863 



91932 
92006 
92063 
92137 
92205 



92274 

92342 
92416 

92478 
92546 



92615 
92683 
92751 
92819 
92887 



92955 
93022 
93096 

93158 
93226 



93361 
93429 
93496 
93564 

8.93631 

iOS. Kxsec. 



70 
70 
70 

69 
70 
70 
69 
70 

69 
70 
69 
69 
69 
69 
69 
69 
69 
69 
69 

69 

69 
69 

69 
69 
69 
69 

68 
69 
69 
68 
69 
68 
68 
68 

69 
68 
68 

68 
68 

68 
68 

68 
68 
68 

68 
68 
68 
68 
68 

68 

67 
68 

67 
68 

67 
68 

67 
67 
67 
67 

"77" 



Lotf. Vers. 



J> 



8 . 90034 
90096 
901 58 
90220 
90282 



90344 
90406 

90467 
90529 

90591 



90652 

90714 
90776 
90837 
90899 



90966 
91021 
91083 

91144 
91205 



91267 
91328 

91389 
91450 
91511 



91572 

91633 
91694 
91755 
91815 



91876 
91937 
91997 

92058 
921 19 



92179 
92240 
92306 
92361 
92421 



92487 

92542 
92602 
92662 
92722 



92782 
92842 
92902 
92962 
93022 



93082 

93142 
93202 
93261 
93321 



93381 
93440 
93506 
93560 
93619 
8.93679 

lidsr. Vers. 



62 
62 
62 
62 

62 
62 
61 
62 
61 

61 
62 
61 
61 
61 
61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 

6r 
61 
66 

61 
66 
66 
61 
66 

66 
66 
66 
66 
60 

66 
66 
60 
60 
66 

60 
60 
60 
60 
60 

60 

59 
60 

59 
60 

59 
59 
60 

59 
59 
59 

/> 



Loir. Kxsec. 



7> 



8 



8 



93631 
93699 
93766 

93833 
93901 



93968 

94035 
94102 

94170 
94237 
94304 
94371 
94438 
94505 
94572 



94638 
94705 
94772 
94839 
94905 



94972 
95039 
95105 
95172 

95238 



95305 
95371 
9543? 
95504 
95570 



95636 
95703 
95769 

95835 
95901 



95967 
96033 
96099 
96165 
96231 



96297 
96362 
96423 

96494 
96560 



96625 
96691 
96757 
96822 
96888 

96953 
97013 
97084 

97149 
97214 

97280 

97345 
97410 

97475 
97540 

()76o''> 

jOir. Kxsec. 



67 
67 
67 
67 
67 
67 
67 
67 
67 

67 
67 
67 
67 
67 

66 
67 
66 
67 
66 

66 
67 
66 
66 
66 

66 
66 
66 

66 
66 
66 

66 
66 
66 
66 

66 
66 
66 
66 
66 

66 

65 
66 

65 
66 

65 

65 
66 

65 
65 

65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 

/> 



10 

1 1 

12 

13 
14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 

i9 
30 

31 
32 
33 

34 



35 
36 

37 
38 

39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 

51 
52 
53 
54 



55 
56 

57 
58 

sQ 
CO 





70 


69 


6 


70 


0.9 


7 


8 


I 


8.0 


8 


9 


3 


9.2 


9 


10 


5 


10.3 


10 


II 


f> 


11. 5 


20 


23 


3 


23.0 


.1° 


35 





34-5 


40 


46 


fi 


46.0 


50 


5B 


3 


57-5 



68 

6.8 

7.9 

9.0 

10.2 

It. 3 

22.6 
34.0 
45-3 

56.0 





67 


66 


6 


6.7 


0.6 


7 


7.8 


7.7 


8 


8.9 


8.8 


9 


10.0 


9.9 


10 


II. I 


II. 


20 


22 3 


22.0 


30 


33.5 


33.0 


40 


44-6 


44.0 


50 


55. 8 


55.0 



65 

6.5 
7.6 

9.7 
10. § 

21.6 
32.5 
43-3 
54.1 





64 


63 


62 


6 


6.4 


6.3 


6. 


7 


7.4 


7-3 


7- 


8 


8.5 


8.4 


8. 


9 


9.6 


9.4 


9- 


10 


10 6 


10.5 


10 


■20 


= 1-3 


21.0 


20. 


30 


32.0 


31.5 


3» 


40 


42.6 


42.0 


41. 


50 


53.3 


52.5 


5'. 





6 


[ 


60 


6 


6.1 


6.0 


7 
8 


7 
8 


I 
I 


7.0 
8.0 


9 


9 


I 


9.0 


10 


10 


I 


10. 


30 


20 


3 


20.0 


30 


30 


5 


30.0 


40 


40 6 


40.0 


50 


50 


8 


50.0 



59 

5-9 
6.9 

H 

9.§ 

»9 6 

29.5 

39.3 
49. i 





5 


6 


0.0 








7 








8 








9 





I 


10 





t 


20 





I 


30 





2 


40 





3 


50 





4 



I'. I". 



405 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

24° 25° 



10 

1 1 

12 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 
27 



29 



30 

31 
32 
33 

34 



35 

36 
37 
38 
39 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



Los;. Vers. 



8 



00 



93679 
93738 
93797 
93857 
93916 



D iLoff. Exsec 



93975 
94034 
94094 

94153 
94212 



94271 
94330 
94389 
94448 
94506 



94565 
94624 
94683 

94742 
94800 



94859 
94917 
94976 

95034 
95093 



95151 
95210 
95268 
95326 
95384 



95443 
95501 

95559 

95617 

9567? 



95733 
95791 
95849 
95907 
95965 



96023 
96086 
96138 
96196 
96253 



9631 1 

96368 
96426 

96483 
96541 



96598 
96656 
96713 
96776 
9682^ 



96885 
96942 

96999 
97056 
97113 



97170 



59 

59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
58 

59 
59 

58 
59 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 

58 
58 
58 
58 

58 

58 

58 
57 
58 
58 
58 
Sf 
SJ 
58 
57 

5f 
57 
57 
5f 
57 

57 
57 
57 
57 
57 

5? 
57 
57 
Si 
57 
57 



8.97606 
.97671 

.97736 
.97801 
•97865 



>. 97930 

•97995 
. 98060 

.98125 
.98190 



I) 



8.98254 
.98319 

.98383 



.98513 



8.98577 
.98642 

.98706 
.98776 

•98835 



8.98899 

.98963 
.99028 
.99092 
•99156 



.99220 
.99284 

■99348 
.99412 

•99476 



;. 99540 
.99604 

.99668 
.99732 
.99796 



8.99860 

.99923 

8.99987 

9.00051 

.001 14 



9.OJ178 
. 00242 
.00305 
,00369 
.00432 



9-00495 
.00559 
.00622 
.00686 
.00749 



Log. Vers. 1 I) Log. Kxsec. J> 



9 . 008 I 2 
.00875 
.00938 
.01002 
.01065 



9.01 128 
.01191 
.01254 
.01317 
.01380 



9.01443 



65 
65 
65 
64 

65 
65 
64 
65 
65 
64 
64 
64 
65 
64 

64 
I 64 

64 
64 
64 
64 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
63 
64 
63 
64 
63 
63 
64 
63 
63 
63 
63 

63 
63 
63 
63 
63 
63 
63 
63 
63 
63 

63 
63 
63 
63 
63 
63 



Log. Vers. 



8.97176 
9722^ 
97284 

97341 
97398 



8 



I) Log. Exsec. 2> 



97455 
97511 

97568 
97625 
9768T 



97738 

97795 
97851 

97908 
97964 



98026 
98077 

98133 
98190 
98246 



98302 

98358 
98414 

98470 
98527 



98583 
98639 
98695 
98750 
98806 



98802 

989 1 8 

98974 
99030 

99085 



99141 
99197 
99252 
99308 
99363 



99419 

99474 
99529 

99585 
99646 



99695 

99751 
99806 
99861 

999 1 6 



99971 
00025 
00081 
00136 
00 1 91 



00246 
00301 

00356 
0041 1 

00466 



9.00520 



Lost. V«»rs. 



57 
56 
57 
57 

57 
56 
57 
56 
56 

56 
57 
56 
56 
56 
56 
56 
56 
56 
56 

56 
56 
56 
56 
56 
56 
56 
56 
55 
56 

56 
56 

55 

56 

55 
55 

56 

55 
55 
55 
55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 

55 
55 
54 
55 
54 



01443 
01505 

01568 
01631 
01694 



OI756 
01819 

01882 

01944 

02007 



02070 
02132 
02195 
02257 
02319 



02382 
02444 
02506 
02569 
02631 



02693 

02755 
0281^ 

02880 

02942 



03004 
03066 
03128 
03190 
03252 



03313 

03375 

03437 

03499 
03561 



03622 
03684 
03746 
03807 
03869 



03930 

03992 

04053 
041 1 5 

04176 



04238 
04299 
04366 
0442 T 
04483 



04544 
04605 
04666 
04727 
04788 



04850 
0491 1 
04972 
05033 
£i°93 
05154 



62 

63 
62 

63 
62 

63 
62 
62 
63 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

61 
62 
62 
61 
62 

61 
61 
62 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
6t 

61 
61 
61 
61 
66 
61 



/> Loir. Exsec. I /> 



5 
6 

7 
8 

9 
10 

II 
12 

13 

14 



25 
26 
27 
28 
29 



30 

31 
32 

33 
34 



35 
36 

37 
38 
39 



40 

41 

42 
43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



00 



i». p. 



65 64 63 



6 


6.5 


6.4 


7 


7.6 


7-4 


8 


8-6 


8^5 


9 


9-7 


9.6 


10 


IO-8 


10.6 


2? 


21-6 


21.3 


30 


32-5 


32.0 


40 


43-3 


42.6 


50 


54-1 


53 3 



40 
50 





56 


55 


6 


5-6 


5^5 


7 


6^5 


6.4 


8 


7-4 


7^3 


9 


8.4 


8.2 


10 


9-3 


9.1 


20 


i«.6 


18.3 


30 


28.0 


27^5 


40 


37-3 


36^6 


50 


46.6 


45-8 



8.4 

9.4 

10.5 

21 .0 

3i^5 
42.0 

52.5 



62 61 60 



6. 

7.0 

8.0 

9^ 

10.0 
20.0 
30.0 
40.0 
50- 





59 


58 


6 


5^9 


5.8 


7 


6.9 


6.7 


8 


7-8 


7-7 


9 


8-8 


8.7 


lO 


9^8 


9-6 


20 


19-6 


19-3 


30 


29^5 


29.0 


40 


39-3 


38.6 


50 


49.1 


48.3 



57 

5-7 
6.6 
7.6 

8.5 
9-5 

19 

28 

38 
47-5 



54 

5.4 

6. 

7- 
8. 

9- 
18. 
97. 
36.0 
45- 



P. P. 



406 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



^G 



27 



liOK. VlMV 



/> 



10 

1 1 

12 

13 
14 



15 
16 

18 
10 



'20 

21 

22 

^3 

^4 



-3 

26 
27 
28 
29 



;{o 

31 
32 
33 

34 



35 
56 

37 
38 
30 



4tl 

41 
42 
43 
44 

45 
46 
47 
48 
49 



9.00520 
.00575 
.00630 
.00684 
.00739 



9.00794 
.00848 
.00903 
.00957 
.0101 T 



9.0 
.0 
.0 
.0 

.0 



9.0 
.0 
.0 
.0 

.0 



9.0 
.0 
.0 

.0 

.0 



9.0 
.0 
.0 



066 
120 

174 
229 
28^, 



337 
391 
44 3 
499 



t)05 

662 

715 
769 
823 



877 

93' 
985 

.02038 
.02092 



9.02146 
.02199 
.02253 
.02307 
.02360 



9.02414 
.02467 
.02521 

.02574 
.02627 



9.02681 
.02734 
.02787 
.02840 
.02894 



9.02947 
.03000 

.03053 
. 03 1 o5 

.03159 



51 

- '^ 

54 



35 
56 
57 
58 
59 
00 



9.03212 
.03265 
.03318 
•03371 
•03423 



9 -03476 
.03529 

.03582 
•03634 
•03^87 
9-03740 

liOa:. Vers. I /> 



55 
54 
5-+ 
54 

55 
54 
54 
54 
54 
54 
5-i 
54 
54 
34 

54 
54 
5 + 
54 
54 

54 
54 
53 
54 
54 

54 
53 
54 
53 
54 

53 
53 
54 
53 
53 
53 

- -> 

53 
53 
53 

53 

53 
53 
53 

53 
53 

53 
53 
53 

53 
53 
52 

53 
53 
52 
52 
53 
52 



liOir. Kxsec 



/> 



9.03154 
•O521S 
•05276 
.05337 
•05398 



9^05458 
.05519 
.05580 
.05640 
.05701 



9.05762 
.05822 
.05883 

•05943 

. 06004 

9 . 06064 
. 06 1 24 
.06185 
.0624^ 
.06305 



9.06366 
.06426 
.06485 
.06546 
. 06605 



9 . 06667 
.06727 
.06787 
.06847 
.06907 



9.06967 
.07027 
.07087 

•07146 
.07205 



9.07265 
.07326 
.07386 
• 074-L^ 
.07505 



9.07565 
.07624 
.07684 

.07743 
.07803 



9.07863 
.07922 
.07981 
. 0804 I 
. 08 1 00 



9 . 08 1 60 
.08219 
•08278 
•08338 
•08397 



9.08455 
.0851I 
.08574 
.08634 
■ 08693 

9.08752 

MH. Kxspc. 



61 
61 

66 

61 
60 
61 

66 
66 
66 

61 
66 
66 
66 
66 
66 
60 
60 
,60 
60 

66 
60 
66 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 

59 
60 

60 

59 
60 

59 
60 

59 

59 

59 

59 
60 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 
59 



I am:. V«ms. 



/> 



9.03740 

.03792 

.03845 
.03898 

.03950 



,04002 

04055 
.04107 
,04160 
,0421 2 



9.04264 

.04317 
.04369 
.O442T 

•04473 



9.0452^ 

.04577 
.04630 
.04682 
.04734 



9.04786 

•04837 
.04889 
.04941 
•04993 



9.05045 
.05097 

•05148 
.05206 
.05252 



9^05303 

•0535^ 
.05407 

•05458 
.05510 



905561 
.05613 
.05664 

.05715 
.05767 



9-05818 
.05869 
.05921 
.05972 
.06023 



9.06074 
. 06 1 2 5 
.06175 
.06227 
.06279 



9.06330 
.06386 
.06431 
.06482 
-06533 



9.06584 
.06635 
.06686 
.06735 
.0678^ 

9.06838 

liOif. Vers. 



52 
52 

53 
52 

52 
52 
52 
52 
52 

52 

52 
52 

52 
52 

52 
52 
52 

52 
52 

52 
51 
52 
52 
52 

51 
52 
51 
52 
52 

51 
52 
51 
51 
51 

51 
51 
5> 
51 
51 

5J 
51 
51 
5' 
51 

51 
51 
51 
51 
51 

5' 
50 
51 
51 
51 

51 
50 
5' 
50 
5' 
50 

/> 



1.0 



l. \^t'(■. 



/> 



08752 I 
0881 I 
08870 
08929 
08988 



09047 
09106 
09164 
09223 
09282 



09341 
09400 

094 5 8 
09517 

09576 



09634 
09693 
09752 
09816 
09869 



09927 

09986 

0044 

0102 

0161 



0219 

0278 

0336 

039-+ 
0452 



051 1 
0569 
0627 
0685 
0743 



080T 

0859 
0917 

097^ 
033 



091 
149 
207 
265 
323 



386 
438 
496 
554 
61T 



669 
727 
784 
842 

899 



95^ 
2015 
2072 
2129 
2 1 87 

2244 

K\s«>r. 



59 
59 
59 
59 

59 
59 
58 
59 

59 

58 
59 
58 
59 
58 

58 
58 
59 
58 
58 
58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
57 
58 
58 

57 
58 
58 
57 
57 
58 
57 
57 
5/ 
57 
58 

3/ 

57 

57 
57 
57 

"77" 



4 

5 
6 

7 
8 

9 
10 

1 1 
12 



J 



15 

16 

17 
18 

19 



20 

21 
22 

23 
24 



25 
26 

27 
28 

^_ 

;jo 

31 
32 
jj 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 

45 
46 

47 
48 

50 

5' 

52 
53 
54 

55 
56 
57 
58 

59 



r. 1' 



6 


6 

6 


I 


60 

6.0 1 


7 
8 


7 
8 




7.0 
8.0 


9 
10 


9 

lO 




9.0 1 
10. 


20 


20 


.3 


20.0 


30 

no 
50 


30 
40 
50 


5 
8 


30 
40.0 
50.0 



59 

5-9 

6.9 

7-8 

8.8 

9-8 

'9-§ 

29.5 

39.3 

49.1 



58 57 



6 


5-8 


7 


6.7 


8 


7-7 


9 


8-7 


10 


9-6 


20 


19.3 


30 


29.0 


40 


3S.6 


50 


48.3 



5-7 

6.6 

7.6 

8.5 

9^5 

19.0 

28.5 

38.0 

47^5 





55 


.54 


6 


5-5 


5^4 


7 


6.4 


6.3 


8 


7-3 


7.2 


9 


8.2 


8.1 


10 


9^1 


9.0 


20 


18.3 


18.0 


30 


27.5 


27.0 


40 


36.^ 


36.0 


50 


45^8 


45.0 





53 


52 


6 


5^3 


5.2 


7 


6.2 


6.0 


8 


7.0 


6.9 


9 


7-9 


7.8 


10 


8^8 


8-6 


20 


'7-6 


17.3 


30 


26.5 


26.0 


40 


35^3 


34. 6 


50 


44.1 


43.3 





51 





6 


5.1 


0.6 


7 


5.9 


0.5 


8 


6.8 


0.0 


9 


7.6 


o.x 


10 


8.5 


O.T 


ao 


17.0 


O.I 


30 


25.8 


0.2 


40 


34.0 


03 


50 


42.=; 


0.4 



I'. p 



407 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

28° 29° 



Log. Vers. I) 






9.06838 


I 


.06888 


2 


.06939 


3 


. 06996 


4 


. 07040 


5 


9.07091 


6 


.07141 


7 


.07192 


8 


.07242 


9 


.07293 


10 


9 -07 343 


II 


.97393 


12 


.07444 


13 


.07494 


14 


■07544 


15 


9.07594 


i6 


.07644 


17 


.07695 


i8 


.07745 


19 


.0779^ 


20 


9.07845 


21 


.07895 


22 


.07945 


23 


.07995 


24 


.08045 


25 


9.08095 


26 


.08145 


27 


.08195 


28 


.08244 


29 


.08294 ^ 


30 


9-Oii344 


31 


.08394 


32 


.08443 


33 


.08493 


34 


•08543 


35 


9.08592 


36 


.08642 


37 


.08691 


3« 


.08741 


39 


.08790 


40 


9.08840 


41 


.08889 


42 


.08939 


43 


.08988 


44 


.090^^7 


45 


9.09087 


46 


.09136 


47 


.09185 


48 


.09234 


49 


.09284 


50 


9.09333 


51 


.09382 


52 


.09431 


53 


. 09480 


54 


.09529 


55 


9-09578 


56 


.0962^ 


57 


•09676 


5^ 


.09725 


59 


.09774 


60 


9.09823 



liOSj. Vers. 



50 
51 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 

49 
50 

49 
50 
49 
49 
50 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 
49 
49 
49 
48 
49 
49 



Loar. Kxsec. ! Z> 



7> 



2244 
2302 
2359 

24 1 6 

2474 



2531 
2588 
2645 

2703 
2760 



2817 

2874 
293T 

2988 

3045 



3102 
3159 
3216 

3273 
33"^o 



33^7 
3444 
3500 

3557 
3614 



3671 
3727 
3784 
3841 
3897 



3954 
401 1 
4067 
4124 
4180 



42J7 
4293 
4350 
4406 
4462 



45^9 

457? 

4631 
4688 

4744 



4800 
4856 

4913 
4969 
5025 



5081 

513? 
5193 
5249 
5305 



5361 
541^ 
5473 
5529 
5585 



5641 



Si 
57 
Si 
57 
Si 
57 
57 
57 

57 
57 
57 
57 
57 
57 
57 
57 
56 
57 

57 
57 
56 
57 
56 
57 
56 
57 
56 
56 
57 
56 
56 
56 
56 
56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 
56 
56 
56 
56 
55 
56 



Loff. Vers. 



9.09823 

09872 

09926 

09969 

0018 



f>Off. Exsec. D Lost. Vers. 



0067 
01 ll 
0164 
0213 
O26T 



0310 

0358 
0407 

0455 
0504 



0552 
0601 
0649 
069^ 
0746 



0794 
0842 
0896 

0939 
0987 



035 
083 

131 
179 

227 



-D Los. Exsec. 



323 
371 
419 

467 



515 

562 
616 

658 

706 



754 
801 

849 
897 
944 



992 
2039 
2087 

2134 
2182 



2229 

2277 

2324 
2371 
2419 



2466 

2513 
2566 

2608 

2655 



2702 



49 
48 
49 
48 

49 
48 
48 
49 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 
48 

48 
48 

48 
48 

48 
48 
48 
48 
48 
48 
48 
48 
Al 
48 

48 

47 
48 

48 
47 
48 
47 
Al 
48 

47 

Al 
Al 
47 
47 
47 

Al 
Al 
47 
47 
47 
47 
Al 
47 
Al 
47 
47 



n 



J) 



5641 
5697 
5752 
5808 
5864 
5920 

5975 
6031 

6087 
6142 



6198 
6254 
6309 
6365 
6426 



6476 

6531 
6587 
6642 
6698 



6753 
6808 
6864 
6919 
6974 



7029 
7085 
7140 

7195 
7256 



7305 
7361 

7416 
7471 
7526 



7581 
7636 
7691 
7746 
78or 



7856 
7916 
7965 
8026 
8075 



8130 
8185 
8239 
8294 
8349 



8403 
8458 
8513 
856^ 
8622 



8676 
8731 
8786 
8846 
8894 



8949 



56 
Si 
56 
55 
56 
Si 
56 
Si 
55 

56 
55 
55 
55 
55 

55 
55 
55 

55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 
55 
55 
55 
55 
55 

55 
54 
55 
55 
54 

55 
55 
54 
55 
54 

54 
55 
54 
54 
54 
54 
54 
55 
54 
54 
54 



TiOii. Exseo. Jt 



20 

21 

22 

23 
24 



25 
26 

27 
28 

29 



p. p. 



30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



(JO 



5l 57 58 



54 

5.4 

6.3 

7.2 

8.1 

9.0 

18.0 

27.0 

36.0 

45.0 



51 50 





54 


6 


5.4 


7 


6.3 


8 


7.2 


9 


8.2 


10 


9.1 


20 


18. £ 


30 


27.2 


40 


36.3 


50 


45-4 



5-7 
6.7 


5-7 
6.6 


6.0 


7-6 
8.6 


7.6 
8.5 


7-5 
8.5 


9.6 


9-5 


9.4 


10. 1 


19.0 


18.8 


28.7 


=8.5 


28.2 


38.3 


38.0 


37-6 


47.9 


47-5 


47.1 





56 


55 


6 


5-6 


5-5 1 


7 


6.5 


6 


5 


8 


7-4 


7 


4 


9 


8.4 


8 


3 


10 


9.3 


9 


2 


20 


18.6 


18 


5 


30 


28.0 


27 


7 


40 


37-3 


37 





50 


46.6 


46 


2 



9.1 



27. 

36. 

45- 



5-1 


5-0 


5- 


5-9 
6.8 


5-9 
6.7 


5- 
6. 


7.6 

8.5 

17.0 


7.6 

8.4 

16.8 


7- 

8. 

16 


25.5 


25.2 


25- 


340 
42.5 


33-6 
42.1 


33 • 
41. 





49 


49 


Al 


6 


4.9 


4.9 


4 


7 
8 


5.8 

6.6 


5-7 
6.5 


5- 
6. 


9 
10 


7-4 
8.2 


7-3 
8i 


7- 
8 


20 


16.5 


16.3 


16. 


30 


24.7 


24-5 


24. 


40 


33.0 


32-6 


32- 


50 


41.2 


40.3 


40. 





48 


4f 


6 


4.8 


4-7 


7 


5-6 


5 5 


8 


6.4 


6-3 


9 


7.2 


7.1 


10 


8.0 


7-9 


20 


16.0 


15-8 


30 


24.0 


23.7 


40 


32.0 


31-6 


50 


40.0 


39-6 



7- 
15- 
23. 

31- 
39- 



P. P 



40a 



TABLF VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

30° 31° 



Loc Vers. 



I) 



2702 
2749 

2796 
2843 
2896 



2937 
2984 
3031 
3078 



3172 
3219 
3266 

3313 

33^9 



3406 

3453 
3500 

3546 
3593 



3(^39 
3686 
3733 
3779 
3826 



3872 

3919 
3965 
401 1 

4058 



4104 
4151 

4197 

4243 

428g 



433(3 
4382 
4428 

4474 
45^0 



45% 
4612 

4658 
4704 

4750 



4796 
4842 

4888 

4934 
4980 



5026 
5071 
5117 

5163 
5209 



5254 
5306 

5346 
5391 
5437 



1483 



Los;. Vers. 



47 
47 
47 
47 

47 
47 
47 
47 

47 

47 
46 
47 
47 
46 
47 
46 
47 
46 
46 

46 
47 
46 
4i, 
46 

46 
46 
46 
46 

46 

46 
46 
46 

46 
46 

46 
46 
46 

46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 

45 
46 

45 
46 
46 
45 
45 

46 

45 
45 
45 

46 



iOJT. Kxsec. 



J> 



9 ■ ' ^949 
19003 
19058 
191 12 
19167 



n 



19221 

19275 
19329 

19384 
19438 



19492 

19546 
1 960 1 
19655 

19709 



19763 
1 9817 
1 987 1 
19925 

1997^^ 



20033 
20087 
20141 
20195 
20249 



20^0 



j^j 



20357 
204 r I 
20465 
20518 



20572 
20626 
20680 

20733 

20787 



20841 
20894 

20948 
21002 
210; = 



21 109 
21162 I 
21216 
21269 
21323 



21376} 
21430 1 

21483; 
21537 
21 596 



21643 
21697 
21750 
21803 

21857 



21910 
21963 
22015 
22070 
22123 



22176 



54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 
54 
54 
54 
53 
54 
53 
54 
53 
54 

53 
53 
54 
53 
52 

53 
53 
53 
53 

53 

53 
53 
53 

53 
53 

53 

53 
53 
53 
53 

53 
53 
53 
53 
53 
53 



Loar. Vei! 



n I. 



I<0K. Kxsec. J> I Loii. Vers 



Q 



54^^; 3 
5528 
5574 
5619 
5665 



5710 

5755 
5801 

5846 
589T 



5937 
5982 
6027 
6073 
6118 



6163 
6208 
6253 
6298 
6343 



6388 
6434 
6479 
6523 
6568 



6613 
6658 
6703 

6748 
6793 



6838 
6882 
692^ 
6972 
7017 



7061 
7106 
7151 

7195 
7240 



7284 
7329 

7373 
7418 
7462 



7507 

7551 
7596 
7640 
7684 



7729 

7773 
7817 
786T 
7906 



7950 
7994 

8038 
8082 

8i26 



8170 



45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 

45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
44 
45 

45 
45 
45 
45 
44 

45 
44 
45 
44 
45 
44 
44 
45 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 



/> 



K\' 



/> 



22170 
2222Q 
22282 

22335 
22388 



22441 
22494 
22547 
22606 
22653 



227O6 

22759 
22812 

22865 

22918 



22971 
23024 

23076 
23129 
23182 



23235 
23287 

23340 

23393 
23446 



23498 I 

23551 
23603 , 

23656 i 

23709 



23761 
23814 
23866 
23919 

23971 



24024 
24076 
24128 
24181 

24233 



24285 

24338 
24396 

24442 

24495 



24547 

24599 
24651 
24704 
24756 



24808 
24860 
24912 
24964 
25016 



25068 
25126 
25172 
25224 
2 527(3 

25328 



L<tir. Kxsec. 



53 
53 

53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
52 

53 
53 
52 
53 
52 

53 
52 
53 

52 
53 
52 
52 
52 
52 
53 
52 
52 
52 
52 
52 

53 
52 
52 
52 
52 

52 
52 
52 
52 
52 
52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 
52 



jt 



4 

5 
6 

7 
8 

_9 
10 

1 1 

I 2 



14 

15 
16 

17 
18 

19 



20 

21 

22 

23 

24 



!'. I' 



54 54 53 



5-4 
63 


5-4 
6.3 


7-2 

8.2 


7.2 
8.1 


9.1 
18.1 


90 
18 


27.2 
36.3 


27.0 
36.0 


45-4 


45. 



5 3 
0.2 

7- 

8 

8.9 

17 8 
26.7 

35-6 
44.6 





S3 


52 


6 
7 


5-3 
6.2 


5-2 


8 


7.0 


7.0 


9 
10 


7-9 
8-8 


7-9 
8.7 


20 

30 


17-6 
26.5 


'7-5 
a6.2 


40 


35-3 


35 -o 


50 


441 


43-7 



52 

5-2 

6.5 
6.9 
7.8 

86 
17-3 
26.0 





47 


47 


6 


4-7 


4-7 


7 


5-5 


5-5 


8 


6.3 


6.2 


9 


71 


7.0 


10 


7-9 


7-8 


20 


'5-8 


15-6 


30 


23.7 


23 -5 


40 


31-6 


31-3 


50 


39-6 


39- 1 



46 

4-6 
5-4 
6.2 
7.0 
7-7 

^5-5 

23- 

31 .0 

38.7" 



46 4S 45 



6 


4.6 


4-5 


7 


5-3 


5-3 


8 


6.1 


6.6 


9 


6.9 


6.8 


10 


7-§ 


7.6 


20 


»5-3 


15. 1 


30 


23.0 


22.7 


40 


3?-6 


30.3 


50 


38.3 


37.9 



4' 

5' 

6. 

6. 

7' 
15' 
22.5 
30. 
37- 



20 
30 
40 
50 



44 

4-4 

5-2 

5-9 

6.7 

7-4 

i4-§ 

22.2 

29-6 
37.1 



44 

4-4 

5-« 

5-8 
6.6 

7-3 
14-6 
22.0 

29-3 
36.6 



v. r. 



409 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT! 

33° 33° 



10 

II 

12 

13 

14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 

25 
26 
27 
28 
29 



Los. A'ers. 1> Lofr. Exsee 



30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 ! 

41 
42 

43 

44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 

57 
58 
59 



(io 



8170 
8214 
8258 
8302 

8346 



8390 
8434 

8478 
8522 
8566 



8610 
8654 
8697 
874t 

878s 



8829 
8872 
8916 
8959 
9003 



9047 
9090 

9134 
9177 
9221 



9264 
9308 

9351 
9395 
9438 



9481 

9525 
9568 
961 T 

9654 



9698 

9741 
9784 
9827 
9870 



9914 

9957 
20000 
20043 
20086 



20129 
20172 
20215 
20258 

2030[ 



20343 
20385 

20429 
20472 

20515 



20558 

20600 
20643 

20686 

20728 



9.20771 



Log. Vers. 



44 
44 
44 
44 
44 
44 
44 
44 
43 
44 
44 
43 
44 
43 
44 
43 
43 
43 
44 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
42 
43 
43 
43 
42 

43 
42 
43 
43 

42 
43 



2> 



25328 
25386 

25432 
25484 
25536 



25588 
25640 
25692 

25743 
25793 



25847 
25899 
25950 
26002 
26054 



26105 
26157 
26209 
26260 
26312 



J) 



Log. V«^rs. 1} 



26364 
26415 
26467 

265I8 
26570 



26621 
26673 
26724 
26776 
26827 



26878 
26930 
2698T 
27032 
27084 



27135 
27185 

27238 
27289 
27340 



27391 

27443 
27494 
27545 

27596 



27647 

27698 

27749 
27800 

27852 



27903 

27954 
28005 
28056 
28107 



28157 
28208 
28259 
28316 
2836T 



28412 



Log. Exsec. 



52 
52 
52 
51 
52 
52 
52 
51 
52 

51 

52 
51 
52 
51 

51 
52 
51 
51 
51 
52 
51 
51 
51 
51 

51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 

51 
51 
51 
51 
51 

51 
51 
51 
51 
51 

51 
51 
51 
51 
51 

50 
51 
51 
51 
51 
50 



7> 



9.20771 
20814 
20855 
20899 
20942 



20984 
027 
069 
112 
154 



196 
239 
281 

324 
-.66 



408 
451 
493 

535 
577 



620 
662 
704 
746 
788 



836 
872 
914 
956 
998 



22040 
22082 
22124 
22165 
22208 



22250 
22292 
22334 
22376 
2241^ 



22459 
22501 

22543 
22584 
22626 



22668 
22709 
22751 
22792 
22834 



22875 
22917 
22959 
23006 
23042 



23083 
23124 
23166 
2320^ 
23248 



9.23290 



Log. Vers. 



42 

42 
42 

43 
42 
42 
42 
42 
42 

42 
42 

42 
42 
42 

42 

42 

42 
42 
ZI2 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 

42 
42 
42 

41 

42 

42 
41 
42 
41 
42 
41 
41 
42 
41 
41 
41 
41 

41 
41 
41 
41 
41 

41 
41 
41 
41 
41 
41 



ILoi 
9 



Exsec. 



I) 



28412 
28463 
28514 
28564 
28615 



28665 
28717 
28768 
28818 
28869 



28920 
28976 
29021 
29072 
29122 



29173 
29223 
29274 
29324 

29375 



29426 

29476 
29527 
29577 
29627 



29678 
29728 

29779 
29829 
29879 



29930 
29986 
30036 
30081 
30131 



3018T 
3023T 
30282 

30332 
30382 



I) 



30432 
30482 

30533 
30583 
30633 



30683 

30733 
30783 
30833 
30883 



30933 
30983 
31033 
31083 

3fT33 



31183 
31233 
31283 
31333 
31383 



31432 



51 
51 
50 
51 

51 
50 
51 
50 
50 

51 
50 
50 
51 
56 

51 
50 
50 
50 
51 
50 
50 
56 
50 
50 

50 
50 
50 
50 

50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 
50 
50 
50 

50 
50 

50 
50 
50 
50 
50 

49 
50 
50 
50 
50 
49 



15 
16 

17 
18 

19 



20 

21 

22, 

23 
24 



25 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 
50 

51 

52 
53 
il 
55 
56 
57 
58 
59 



'Log. Exseo.l 7> 



GO 



p. p. 





52 


51 


6 


5-2 


5-1 


7 


6.0 


6.0 


8 


6.9 


6-8 


9 


7.8 


7-7 


10 


8.6 


8.6 


20 


17-3 


17. 1 


30 


26.0 


25-7 


40 


34.6 


34-3 


50 


43.3 


42.9 



6 


50 

5-0 


50 

5-0 


7 
8 


5-9 
6.7 


5-8 
6.6 


9 
10 
20 


7.6 

8.4 

16.8 


7-5 
16.6 


30 


25.2 


25.0 


40 


33-6 


33-3 


50 


42.1 


41-6 



6 


44 

4.4 


43 

4-3 


7 
8 

9 


5-1 

5-8 
6.6 


51 
5-8 
6.5 


10 


7-3 


7.2 


20 


M.6 


14-5 


30 


22.0 


21.7 


40 
50 


29-3 
36.6 


29.0 
36.2 



6 


42 

4.2 


42 

4.2 


7 
8 

9 


4.9 

5-6 
6.4 


4.9 
5-6 
6.3 


10 


7-1 


7.0 


20 


14.1 


14.0 


30 
40 


21 .2 
28.3 


21.0 

28.0 


50 


35-4 


35-0 



6 


41 

4.1 


7 
8 


4.S 
5-4 


9 
10 


6.1 

6.8 


20 


J3-6 


30 


20.5 


40 


27-3 


50 


34-1 



P. p. 



410 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

34° 85° 



Vers. 



7> 



23290 

23331 
23372 

23414 
23455 



23496 

23537 

23579 
23620 

23661 



23702 

23743 
237^4 
2382^ 

23866 



23907 

23948 
23989 
24036 
24071 



24112 

24153 
24194 

24235 

24275 



243 1 6 
24357 
24398 

24438 
24479 



24520 
24561 
2460 T 
24642 
24682 



24723 
24764 
24804 
24845 
24885 



24926 

24966 
25007 

25047 

25087 



25128 
25168 
25209 

25249 
25289 



25329 
25370 
25410 
25450 
25496 



25531 
25571 
25611 
25651 
2569T 

9-25731 

Lost. Vers. 



41 
41 
41 

41 

41 
41 
41 
41 
41 

41 

41 
41 

41 
41 

41 
41 
41 
41 
41 

40 
41 
41 
41 
46 

41 
46 

41 
46 

41 
40 
41 
46 
46 
46 

41 
46 
46 
40 
40 
40 

40 
46 

40 
40 

46 
46 

40 
40 

40 
40 
46 
40 
46 
40 

46 
40 
40 
40 

40 
40 

"77" 



li<);r. Kxsec 



n 



432 
482 

532 
582 
6^2 



681 

731 
781 

83' 
886 



930 
980 
32029 
32079 
32129 



32178 
32228 
32277 
32327 

32377 



32426 
32476 

32525 

32575 
32624 



32673 
32723 
32772 
32822 
32871 



32920 
32970 

33019 
33069 
33118 



33167 

33216 
33266 

33315 
33364 



33413 
33463 
33512 
33561 
33616 



33659 
33708 
33758 
33807 
33856 



33905 
33954 
34003 

34052 
34101 



34150 
34199 
34248 
34297 
34346 

34395 



50 
50 
49 
50 

49 
50 
49 
50 
49 
50 
49 
49 
49 
50 

49 
49 
49 
49 
50 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 
49 



Lost. Kxspo. /> 



Lour. Vers. 



25731 
25771 
2581I 
25851 
25891 



25931 
25971 

2601 T 
26051 
26091 



261 31 
26171 
26216 
26256 
26296 



26330 
26370 
26409 
26449 
26489 



26528 
2656^ 
26608 
2664^ 
26687 



26725 
26765 
26806 
26845 
26885 



26924 
26964 
27003 
27042 
27082 



27121 
27161 
27200 
27239 

27278 



27318 
27357 
27396 
2743S 
27475 



27514 

27553 
27592 

27631 
27676 



27709 

27749 
27788 

27827 

27866 



27905 
27944 
27982 
28021 
28066 

28099 

Lost. Vers. 



n 



40 
40 
40 
40 

40 
40 
40 

39 
40 

40 
40 

39 
40 
40 

39 
40 

39 
40 

39 

39 
40 

39 
39 
39 

39 
40 

39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
38 
39 
39 
39 

IT 



\A»i. K.Vht'C. 



It 



34395 
34444 
34492 
3454' 
34590 

34639 
34688 

34737 
34785 
34834 



34883 
34932 
34986 
35029 
35078 

35127 
35175 
35224 
35273 
35321 



35370 
35419 
35467 
35516 
35564 



35613 
35661 

35710 
35758 
35807 



35855 
35904 
35952 
36001 
36049 



36098 

36146 
36194 

36243 
36291 



36340 
36388 

36436 
36484 

36533 
3658T 
36629 
36678 
36726 
36774 



36822 
36876 
36919 
36967 
37015 



37063 
37111 

37159 
3720^ 

37255 
37303 

I,nir. Kxser. 



49 
48 
49 
49 
49 
48 
49 
48 
49 
49 
48 
48 
49 
48 

49 
48 
49 
48 
48 

48 
49 
48 
48 
48 

48 

48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 

48 
48 
48 

48 
48 

48 
48 
48 
48 

48 
48 
48 
48 
48 

77" 



10 



20 

21 
22 

23 
24 



25 
26 
27 
28 
2g 

30 

31 

32 
33 
34 

35 
36 

37 
38 

39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

:>o 

5' 
52 
53 
54 

55 
56 
57 
58 



r. I'. 



6 


50 

5.0 


49 

4.9 


7 

8 


6 6 


5-8 
6.6 


9 
10 


7-5 
8.3 


7-4 
8 2 


20 


16.6 


.6.5 


30 


25.0 


247 


40 


33-3 


33-0 


50 


41 6 


41.2 



40 
50 



20 
30 

40 
50 



40 
50 



20 

30 
40 
50 



48 

4.§ 

5-6 
6.4 
7-3 
8.1 
16. 1 
24.2 

3* -3 
40.4 



41 

4.1 

4-S 
5-5 
6.2 
6.9 

20.7 

27-6 
34.6 



39 

3-9 
4.6 

5-2 
.S-9 
6.6 
13.1 
19.7 
26.3 
32.9 



6 
7 
8 

9 
10 

20 

30 
40 

5'3 



49 

4.9 

5-7 
6.5 

7-1 
8. 

16.3 

24-5 

32.^ 

40.8 



48 

4.8 

5.6 

6.4 

7.2 

8.0 

j6.o 

24 o 

32.0 

40.0 



41 

4.1 
4.8 
5-4 

e.i 

H 

13-6 

20. T 
27.3 

34-1 



40 40 



4 





4- 


4-7 


4- 


5 
6 


4 
I 


5- 
6. 


6 


7 


6. 


13 


5 


'3- 


20 


2 


20. 


27 





26. 


33 


7 


33 



39 

3-9 
4-5 
5-2 
5-8 

13.0 
19.5 
26.0 
32.5 



38 

3-8 

4-5 

51 

S.8 

6.4 

12. g 

19.2 

25-6 

32-1 



I'. I'. 



411 



TABLE VIII.-LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 

36° 37° 



'_ j Log. Vers. I J> JLog. Exseo 



10 

II 

12 

H 



15 
i6 

17 
i8 

19 



20 

21 
22 

23 
24 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 
40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 

56 
57 
58 
59 



60 



9 . 28099 
.28138 
.28177 
.28216 
.28255 



9.28293 
.28332 
.28371 
.28410 
. 28448 



9.28487 
.28526 
.28564 
.28603 
.28642 



9.28680 
.28719 

.2875; 
.28796 

.28835 



9-28873 
.28912 
.28950 
.28988 
.29027 



9.2906^ 
.29104 
.29142 
.29180 
.29219 



9.29257 

.29295 
•29334 
.29372 
.29410 



9-29448 
• 29487 
.29525 

.29563 
. 29601 



9.29639 
.29677 
.2971^ 

•29754 
.29792 



9.29830 
.29868 
. 29906 
.29944 
.29982 



9.30020 
.30057 

•30095 
.30133 
.30171 



9 . 30209 

. 30247 
.30285 
.30322 
■ 30360 



9 -30398 

Log. Vers. 



39 
38 
39 
39 

38 
39 
38 
39 
38 

39 
38 
38 
39 
38 
38 
38 
38 
38 
39 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
3f 
38 
38 
38 
38 
37 
38 
3? 
38 
38 

IT' 



9-37303 
•37352 

• 37400 
.37448 

• 37496 



37544 
37592 
37640 
37687 
37735 



n 



37783 
37831 
37879 
37927 
37975 



38023 
38071 
38119 

38166 
38214 



38262 
38310 

3835^ 
38405 
38453 



38501 
38548 
38596 
38644 
38692 



38739 
38787 

38834 
38882 

38930 



38977 
39025 
39072 
39120 
39168 



39215 
39263 
39310 
39358 
39405 



39453 
39506 

39548 

39595 
39642 



39690 

39737 
39785 
39832 
39879 



39927 

39974 
40021 

40069 

401 16 



40163 



48 
48 

48 
48 

48 
48 
48 

47 
48 

48 
48 
48 

48 
47 
48 
48 
48 

4? 
48 

4^ 
48 
4f 
48 
47 
48 

47 
48 

47 
48 

47 
47 
47 
48 
47 

47 
47 
47 
48 
47 

47 
48 

47 
47 
47 

47 
47 
4? 
47 
47 

4f 
47 
47 
47 
4? 

Al 
47 
4l 
4l 
47 
41 



Loff. Kxsec.l /> 



Log. Vers. I U Log. Exsec.l 2> 



9-30398 

• 30436 

• 30474 
.30511 

• 30549 



9-30587 
. 30624 
. 30662 
. 30700 

-3073? 



9-30775 
.30812 

.30850 

. 30887 

-30925 



9.30962 
.31000 
-31037 

•31075 
.31112 



9.31150 
.31187 
.31224 
.31262 
.31299 



9-31336 
-31374 
.31411 

•31448 
•31485 



9.31523 
.31560 

•31597 
•31634 
.31671 



9-317O8 
•31746 

.31783 
,31820 

-31857 



9.31894 

•31931 
.31968 
.32005 
.32042 



9-32079 
.32116 

.32153 
.32190 

-22227 



9.32263 
.32300 
•32337 
•32374 
.32411 



9-32447 
.32484 
.32521 
•32558 
.32594 

9-32631 
Loe. Vers. 



37 
38 
37 
37 
38 
3l 
3l 
38 
37 

31 
37 
3l 
37 
37 

3l 
37 
37 
3l 
3l 

3l 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

36 

37 
37 
36 
37 

36 

37 
36 
37 
36 

37 

"77" 



9.40163 
.40216 
.40258 
• 40305 
.40352 



• 40399 
,40447 
,40494 
,40541 
40588 



9-40635 
.40682 
•40730 

•40777 
. 40824 



9.40871 

•40918 
.40965 



9-4 
•4 
•4 
.4 
•4 



9.4 
•4 
•4 
-4 

•4 



9-4 
•4 
•4 
•4 
•4 



9-4 
.4 
.4 
•4 
•4 



012 
059 



106 

153 
206 

24^ 
294 



341 

388 

435 
482 

529 



576 
623 
670 

717 
763 



816 

857 

904 

951 
998 



9.42044 
.42091 
.42138 
.42185 

-42231 



9-42278 
-42325 
.42372 
.42418 
.42465 



9.42512 

-42558 
.42605 
.42652 
.42698 



9-42745 
.42792 

.42838 
.42885 

-42931 
9-42978 

Loar. Kxsec. /> 



47 
41 
47 
4l 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
4/ 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

46 
47 
47 
47 
46 
47 
47 
46 
47 
47 

46 
47 
46 
47 
46 
47 
46 
47 
46 
47 

46 
46 
47 
46 
46 

46 
47 
46 
46 
46 
46 



10 



20 

21 

22 

23 
24 



35 
36 
37 
38 
39 



40 

41 
42 
43 

44 



45 
46 
47 
48 
49 



50 

51 
52 

53 
54 



55 
56 
57 
58 
59 
60 



20 

50 



6 

7 
8 

9 
10 

20 
30 
40 

50 



6 

7 
8 

9 
10 
20 
30 
40 
50 



40 

50 



P. P. 



48 

4.8 1 


5 

6 


6 
4 


7 
8 


3 

1 


16 


1 


24 


2 


32 


3 


40 


4 



48 



4^ 



6 

7 
8 


4-7 

5-5 
6.3 


9 


7-1 


10 


7-9 


20 


15. § 


30 


23-7 


40 


31-6 


50 


39-6 



16. 
24. 
32. 
40. 



47 

4-7 

5-5 

6.; 
7-0 

7- 

15.6 
23-5 

31-3 
39- 



20 

30 
40 

50 



46 

4-6 

5-4 

6.2 

7.0 

7-7 

15.5 

23.2 

31.0 

38.7 



39 



38 



3-9 


3- 


4-5 


4. 


5.2 


5- 


5-8 


5; . 


6.5 


6. 


13.0 


12. 


19^5 


19. 


26.0 


25- 


32.5 


32 • 



38 



Zl 



3.8 

4.4 


3 
4 


50 


5 


5-7 
6.3 


5- 
6. 


12.6 


12. 


19.0 


18. 


25-3 


25- 


31-6 


31 



37 



36 



3-7 


3 


4-3 


4- 


4-9 


4- 


5-5 


5- 


6.1 


6. 


12.3 


12. 


18.5 


18. 


24-6 


24. 


30-8 


30- 



P. p 



412 



TABLE VIII,— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 







10 

II 

12 

14 



15 
i6 

i8 
19 



20 

21 

22 

^3 

24 



25 

26 
27 
28 
29 



80 



33 
34 



33 
36 

37 
38 
39 



40 

41 
42 
43 
44_ 

45 
46 

47 
48 
49 



50 

51 
52 
53 
54 



56 
57 
58 
59 
GO 



:58' 



:5i> 



Lost. Vers. /> |li(»sr. Kxsec /> || liOir. Vers, /> liOC. Kxsec 



9.32631 
32668 
32704 

32741 
32778 



32814 
32851 
32888 
32924 
32961 



32997 
33034 
33070 
33107 
33143 



33180 
33216 
33252 
33289 
33325 



33361 
33398 

33434 
33470 
33507 



33543 
33579 
33615 
33652 

33688 



33724 
33766 

33796 
33833 
33869 



33905 
33941 
33977 
34013 
34049 



34085 

34121 

34157 

34193 
34229 



34265 
34301 
34337 
34373 
34408 



34444 
34480 
345 '6 
34552 
34587 



34623 
34659 
34695 
34736 
34766 
9 34802 

Log. Vers. 



36 
36 
37 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 

36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 

36 
36 

36 
36 
36 

36 
35 
36 
36 
3l 
36 
3d 
36 
3l 
36 
35 
36 
35 

/> 



42978 
43024 

43071 
43118 

43164 



432 1 1 

4325^ 
43304 
43356 
43396 



43443 
43489 
43536 
43582 
43629 



43675 
43721 
43768 

43814 
43861 



43907 
43953 
43999 
44046 
44092 



44138 
44185 

44231 

44277 
44323 



44370 
44416 
44462 
44508 

44554 



44601 
44647 
44693 
44739 
44785 



44831 
44877 
44924 

44970 
45016 



45062 
45108 

45154 
45200 
45246 



45292 
45338 
45384 
45430 
45476 



45522 
45568 
45614 
45660 
45706 

9-45752 

iOC. Kxsec. 



46 
47 
46 
46 
46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 

46 
46 
46 
46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 

46 
46 
46 

46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 
46 

"77" 



9.34802 
3483? 
34873 
34909 
34944 



34980 
35016 

35051 
35087 
35122 



35158 

35193 
35229 

35264 
35300 



35335 
35376 
35406 

35441 

35477 



35512 

35547 

35583 
35618 

35653 



35689 
35724 
35759 
35794 
35829 



35865 
35900 

35935 
35976 
36005 



36046 
36076 
361 II 
36146 
36181 



36216 
36251 
36286 
36321 
36356 



3639' 
36426 
36461 

36495 
36536 



36565 
36606 

36635 
36670 

36705 



36739 
36774 
36809 

36844 
36878 

9 • 369 ' 3 



3i) 
36 
35 
35 
35 

36 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

35 
35 
35 
34 
35 

35 
35 
34 
35 
35 
34 
35 
34 
35 
34 
35 



Vers. /> 



It 



9 



45752 

4579^ 
45843 
45889 
45935 



45981 

46027 

46073 
461 18 
46164 



46216 
46256 
46302 
4634^ 
46393 



46439 
46485 

46536 

46576 
46622 



46668 
46713 

46759 
46805 
46856 



46896 
46942 
4698? 
47033 
47078 



47124 
47170 

47215 
47261 

47306 



47352 
47398 
47443 
47489 

47534 



47580 
47625 
47671 

477 '6 
47762 



47807 
47852 
47898 
47943 
47989 



48034 
48080 
48125 
48176 
48216 



48261 
48306 
48352 
4839? 
48442 

48488 

l,llir. K\K4T. 



45 

46 
46 
46 

45 

46 
46 

45 
46 
46 

45 
46 
45 

46 

45 

46 

45 

46 

45 

46 

45 

45 

46 

45 

45 

46 

45 
45 
45 

46 

45 
45 
45 
45 

46 

45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45' 

/> 



5 

6 

7 
8 

9 
10 

1 1 

12 

13 

ii_ 

15 
16 

17 
18 

19 



•20 

21 
22 

23 

24 



25 
26 

27 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 
50 

5' 

52 

53 

J4 

55 
56 
57 
58 

4;o 



V. V 



20 

30 
40 

50 



20 
30 
40 
50 



20 

30 
40 
50 



40 
50 



47 



4-7 
5vS 
6.2 
7.0 
7-8 
15-6 
23 -5 

39- 1 



46 

4.6 

5-1 
6.1 

6.9 

7-6 

15-3 
23.0 

30-6 
38.3 



46 

4-6 

5-4 

6.2 

7.0 

7-7 

15-5 

23.2 

31.0 

38.7 



4S 



4-5 
5-3 
6.0 
6.8 
7.6 
J51 
22.7 

30-3 
37-9 



9 
10 
20 
30 
40 
50 



45 

4-5 

5-2 

6.0 

6.7 

7-5 

150 

22.5 

30.0 

37-5 



37 

3-7 



4-3 

4-9 

5-5 

6.1 

12.3 

18.5 

24-6 

30-8 



36 

3-6 
4.2 

4-8 
5-5 
6.1 
12.1 
18.2 
24.3 
30-4 



36 


35 


3-6 


3-5 


42 


4.1 


4.8 


4-7 


5-4 


5-3 


e.o 


5-0 


J2.0 


"•§ 


18.0 


17-7 


24.0 


23 6 


30.0 


29.6 



35 



34 



6 


3 5 


3-4 


7 


4.1 


4.0 


8 


4-6 


4.6 


9 


5-2 


5-2 


10 


5-§ 


5-7 


20 


n-6 


1 1. 5 


30 


'7-5 


17.2 


40 


23.3 


23.0 


50 


29.1 


28.7 



V. V 



413 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

40° 41° 



Log. Vers. | J> |Log. Exsec. Z> \ Log. Vers. 



10 

II 

12 

13 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 

27 



30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 

42 
43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 
54 



55 
56 

57 
^8 

59 



GO 



36913 
36948 
36982 

37017 
37052 



37086 
3712T 

37156 
37196 

37225 



37259 
37294 
37328 
37363 
37397 



37432 
37466 
37501 
37535 
37570 



37604 
37639 
37673 
377of 
37742 



37776 
37816 

37845 
37879 
37913 



37947 
37982 
38016 
38056 
38084 



38118 
38153 
38187 
38221 

38255 



38289 
38323 
38357 
38391 
38425 



38459 
38493 
38527 
3856T 

38595 



38629 
38663 
38697 

38731 
38765 



38799 
38833 
38866 
38906 

38934 



9.38968 



Loff. Vers. 



34 
34 
35 
34 

34 
35 
34 
34 

34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 
34 
34 
34 
34 
34 

34 
34 
34 
34 
34 
34 
34 
34 
34 
34 

34 
34 
34 
34 
34 
34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
33 
34 

34 
34 

33 
34 
33 
34 



/> 



48488 
48533 

48578 
48624 
48669 



48714 

48759 
48805 
48850 
48895 



48946 
48986 

49031 
49076 
4912T 



49166 
492 1 T 

49257 
49302 

49347 



49392 
49437 
49482 

49527 
49572 



49618 
49663 
49708 

49753 
49798 



49843 
49888 

49933 
49978 
50023 



50068 
50113 
50158 
50203 
50248 



50293 
50338 

50383 
50427 

50472 



50517 
50562 
50607 
50652 
50697 



50742 

50787 
5083T 

50876 
5092T 



50966 
5101 1 

51055 
51 106 

51145 



Q . 5 II 90 



45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 

45 
45 

45 
45 

45 
45 
45 
45 
45 

45 
45 
45 

44 
45 
45 
45 
45 
44 
45 

45 
45 
44 
45 
45 
44 
45 
44 
45 
45 
44 



Log. Hxsec 



9.38968 
39002 

39035 
39069 

39103 



39137 
39176 

39204 
39238 
39271 



39305 

39339 
39372 
39406 

39439 



39473 
39507 
39540 
39574 
3960^ 



39641 
39674 
39708 

39741 
39774 



39808 
39841 
39875 
39908 
39941 



39975 
40008 
40041 
40075 
40108 



40 1 41 
40175 
40208 
4024T 

40274 



40307 
40341 
40374 
40407 
40446 



40473 
40506 
40540 

40573 
40606 



40639 
40672 
40705 

40738 
40771 



40804 
40837 
40870 
40903 
40936 



40969 



U Loar. Exsec 



34 
33 
34 
33 
34 
33 
33 
34 
33 

33 
34 
33 
33 
33 

33 
34 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 

33 

33 

33 

33 

33 

33 
33 
33 

33 
33 
33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 
33 



9 



/> LO!.'. Vers. /> l.n 



190 

235 
279 
324 
369 



n 



414 

458 
503 
548 
592 



63? 
682 

726 
771 
816 



866 

905 
950 
994 
52039 



52084 
52128 

52173 
52217 
52262 



52306 
52351 
52396 
52446 

52485 



52529 
52574 
52618 
52663 

52707 



52752 

52796 
52841 
52885 
52930 



52974 
53018 
53063 
53107 
53152 



53 '96 
53240 
53285 
53329 
53374 



53418 
53462 
53507 
53551 
53595 



53640 
53684 
53728 
53773 
53817 



53861 



45 
44 
45 
44 

45 
44 
45 
44 

44 

45 
44 
44 
45 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 
44 
44 
45 
44 
44 
44 

44 
44 
44 
44 

44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 



Exs"-.. 



/> 



P. P. 



10 

II 

12 

13 
14 



15 
16 

17 
18 

19 



20 

21 

22 

23 

24 



25 
26 
27 
28 

29 
'60 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



50 

51 
52 
53 
ii_ 
55 
56 
57 
58 
59 



({0 



2D 

30 

40 

50 



40 



40 
50 



44 

4.4 

5-2 

5-9 
6.7 

7-4 

14.8 
22.2 

29-6 
37-1 



35 



7 


4 


8 


4 


9 


5 


10 


5 


20 


II 


30 


17 


40 


23 


50 


29 



34 



20 

40 
50 



4S 


45 


4-5 


4-5 


5 • 1 


5-2 


6.0 


6.0 


6.8 


6.7 


7.6 


7-5 


15. 1 


15.0 


22.7 


22.5 


303 


30.0 


37-9 


37-5 



44 

4.4 

5-8 
6.6 

7-3 
14-6 
22 .0 

29-3 
36 6 



34 



33 

3-3 
3-9 
4.4 

50 
5.6 



t6. 



27 



33 



27 



p. P. 



414 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 

4*e" 4;r 



Log. Vers. | 1> 



9 



40969 



001 

034 
067 
106 



133 
166 
199 
231 
264 



297 

330 
362 

395 
428 



461 

493 
525 
559 
591 



624 

657 
689 

722 
754 



7^7 
819 
852 
885 
917 



950 
982 
42014 
42047 
42079 



421 12 
42144 
42177 
42209 
4224T 



42274 
42306 
42338 
42371 
42403 



42435 
42467 
42500 

42532 
42564 



42596 
42629 
42661 
42693 
42725 



42757 
42789 
42822 
42854 
42886 

9-4^9'8 

Loc. Vers. 



32 

33 
jj 
33 
32 
33 
33 
32 
33 
32 
33 
32 
33 
32 

33 
32 
33 
32 
32 
32 
33 
32 
32 
32 

32 
32 
32 
33 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 

7> 



liOj;. KxHec. /> 



53861 
53906 
53950 

53994 
54038 



54083 
54'27 
54171 
54215 
54259 



54304 
54348 
54392 
54436 
54480 



54525 
54569 
546-13 

54657 
5470T 



54745 
54790 
54834 
54878 
54922 



54966 
55016 

55054 
55098 
55142 



55186 
55230 
55275 
55319 
55363 



55407 
55451 
55495 
55539 
55583 



55627 
55671 
55715 
55759 
55803 



55847 
55890 
55934 

55978 
56022 



56065 
561 16 
56154 
56198 
56242 



56286 
56330 
56374 
5641^ 
5646T 

56505 



44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 

44 

44 
44 
44 
44 

44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 

44 

44 
44 
44 
44 
44 

44 
43 
44 
44 
44 

44 
44 
44 
43 
44 

44 
44 
44 
43 
44 
43 



Los:. Kxseo. J> 



Lotr. Vers. 



9 



42918 

42950 
42982 

43014 
43046 



J> 



43078 
43116 

43142 

43174 
43206 



43238 
43270 
43302 
43334 
43365 



4339? 
43429 
4346T 
43493 
43525 



43557 
43588 
43626 

43652 
43684 



43715 
43747 

43779 
43816 

43842 



43874 
43906 
43937 

43969 
44006 



44032 
44064 

44095 
44127 

44' 58 



44190 
44221 

44253 
44284 

443 '6 



44347 

44379 
44416 

44442 

44473 



44504 
44536 
44567 
44599 
44630 



4466 T 

44693 
44724 
44755 
44787 
448 1 8 
L(»e. Vers. 



32 
32 
32 



32 
31 
32 



32 
32 
31 
32 
32 
32 
31 
32 

32 
31 
32 
31 
32 

31 

32 
31 

y 
32 

31 
32 
31 
31 
31 
32 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 

3' 
31 
31 



Loir. K.vNi'c' /> 



W 



56505 

5^>549 
56593 
56637 
56686 

56724 

56768 
56812 
56856 
56899 

56943 
56987 
57031 
57075 
57'J8 
57162 
57206 
57250 
57293 
57337 



57381 
57424 
57468 
57512 

57556 



57599 
57643 
57687 
57730 
57774 



57818 
5786T 
57905 
57949 
57992 



58036 

58079 
58123 
58167 
58216 



58254 

5829? 

58341 

58385 

58428 

58472 

58515 

58559 
58602 

58646 



58689 
58733 
58776 
58826 
58S64 



5890? 
58951 

58994 
59037 
59081 

L'iiii 

y> Ldir. Kxser. 



43 
44 
44 
43 
44 
44 
43 
44 
43 
44 
44 
43 
44 
43 
44 
43 
44 
43 
44 

43 
43 
44 
43 

44 

43 
43 
44 
43 
43 
44 
43 
43 
44 
43 

43 
43 
44 
43 
43 

43 
43 
44 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
44 
43 

43 

43 
43 
43 
43 
43 

/> 



10 

1 I 
12 

13 

_LL 

'5 
16 

17 
18 

19 



20 

21 

-J 

24 



25 

26 
27 
28 

30 

31 
32 
33 
34 

35 
36 

37 
38 
39 



40 

41 

42 
43 

45 
46 
47 
48 

50 

31 
5^ 
53 
54 

55 
5^^ 
57 
58 
59 

<;o 



1'. I*. 



40 
50 



40 
50 



20 
30 
40 
50 



40 
50 



44 



4-4 1 


5 


2 


5 


9 


6 


7 


7 


4 


14 


iJ 


22 




29 


6 


37 


I 



33 



4.4 
4.9 

5-5 
11 .0 
16.5 



27-5 



32 

3-2 
3-7 
4.2 
4.8 

5-3 
10.6 
16.0 
21 3 
26.6 



7 
8 

9 
10 

30 

30 
40 
50 



44 

4.4 

5-i 
5-8 
6.6 

7-3 
14-6 
22 o 

29-3 
36.6 



43 


4 


3 


5 


I 


5 


8 


6 


5 


7 


2 


J4 


5 


21 


7 


29 





36 


2 



43 



28.6 

35-8 



32 



3 

4- 

4- 

5- 
10. 

16. 



27.1 



31 

31 
3-7 
4.7 

4-7 

5-2 

10.5 

»5-7 
21 .0 
26.2 



31 

3-» 
3-6 
4« 

4-6 

51 

10.3 

»5 5 

20.^ 

25-8 



'. I*. 



415 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

44° 45° 



Log. Vers. 






9.44818 


I 


.44849 


2 


.44880 


3 


.44912 


4 


.44943 


5 


9-44974 


6 


.45.005 


7 


•45036 


8 


.45068 


9 


.45099 


10 


9.45130 


II 


.45161 


12 


.45192 


13 


.45223 


14 


.45254 


15 


9.45285 


i6 


.45316 


17 


.45348 


i8 


.45379 


19 


.45410 


20 


9.45441 


21 


.45472 


22 


•45503 


23 


•45534 


24 


.45565 


25 


9.45595 


26 


.4562^ 


27 


•4565? 


28 


.45688 


29 


.45719 


30 


9.45750 


31 


.45781 


32 


.45812 


33 


.45843 


34 


.45873 


35 


9.45904 


36 


.4593! 


37 


•45966 


3« 


•45997 


39 


.46027 


40 


9.46058 


41 


. 46089 


42 


.46120 


43 


.46150 


44 


.46181 


45 


9.46212 


46 


.46242 


47 


.46273 


48 


.46304 


49 


.46334 


50 


9.46365 


51 


.46396 


52 


.46426 


53 


.46457 


54 


.4648^ 


55 


9-46518 


56 


.46549 


57 


.46579 


58 


.46610 


59 


. 46646 


GO 


9.46671 



Z> Log. Ex sec. J> 



Log. Vers. 



31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 
30 
31 
31 
31 
31 

31 
30 
31 
31 
30 

31 
31 
30 
31 
30 

31 
30 
31 
30 
31 
30 
30 
31 
30 
30 

31 
30 
30 
30 
30 

31 
30 
30 
30 
30 
30 

IT 



I. 59124 
.59168 
.59211 

.59255 
• 59298 



59342 
59385 
59429 
59472 

59515 



59559 
59602 

59646 
59689 
59732 



59776 
59819 
59863 
59906 

59949 



9-59993 
. 60036 
.60079 
.60123 
.60166 



, 60209 
,60253 
,60296 
.60339 
.60383 



, 60426 
, 60469 
,60512 
,60556 
60599 



, 60642 
,60685 
,60729 
,60772 
,60815 



,60858 
, 60902 
60945 
,60988 
,61031 



.61075 
,61118 
.61 161 
,61204 
.61247 



,61291 
.61334 
61377 
,61426 
.61463 



9- 



61506 
61550 

•61593 
.61636 

.61679 
9.61722 

Log. Kxsec. I I) 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 

43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 



Log. Vers. 



9 . 4667 1 
4670T 
46732 
46762 
46793 



46823 

46853 
46884 
46914 
46945 



46975 
47005 

47036 

47066 

47096 



47127 

4715^ 
4718^ 
47218 
47248 



47278 
47308 
47339 
47369 
47399 



47429 

47459 
47490 

47520 

47550 



n 



47586 
47616 
47646 
47676 
47706 



47731 
47761 

47791 
47821 

47851 



47881 
4791 1 
47941 
47971 
48001 



48031 
48061 
48096 
48126 
48156 



48186 
48216 
48240 
48270 
48300 



48329 
48359 
48389 
48419 

48449 

48478 



30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

36 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
29 



30 

30 
29 

30 
30 
29 
30 
30 
29 

30 
29 



Log. Vers. I J> 



Log. Exsec. 



9.61722 
.61765 
.61808 
.61852 
.61895 



61938 
,61981 
.62024 
.6206^ 
.621 16 



62153 
.62196 
.62239 
.62282 
,62326 



.62369 
.62412 
.62455 
.62498 
,62541 



9- 



62584 
62627 
62670 
62713 
62756 



9- 



62799 
62842 
62885 
62928 
62971 



63014 
,63057 
,63100 

63143 
63186 



,63229 
.63272 

•63315 
63358 
63401 



9- 



63443 
63486 
,63529 

63572 
63615 



•63658 
•63701 

•63744 
,63787 
,63830 



,63873 

■63915 
•63958 
,64001 
64044 



9.64087 
. 64 1 30 

•64173 
.64216 

•64258 

9-64301^ 

Log. Exsec. J /> 



J) 



43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
42 
43 
43 
43 
43 

43 
42 
43 
43 
43 

43 
42 
43 
43 
43 
42 
43 
43 
43 
42 
43 







10 

1 1 

12 

13 

14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 

27 

28 

29 



30 

31 
32 

34 



35 
36 
37 
38 
39 



40 

41 

42 
43 
44 



45 
46 
47 
48 
49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 
00 



p. P. 



20 

40 
50 



40 

50 



40 
50 



43 

4-3 
5-1 
5.8 
6.5 
7.2 

14-5 



29. 
36. 



6 

7 
8 

9 
10 
20 

30 
40 

50 



31 

3^i 
3-7 
4.2 
4.7 
5-2 
10.5 

15-7 
21 .0 
26.2 



30 

30 

3-5 
4.0 
4.6 

5-1 
10. 1 
15.5 
20.3 
25.4 



20 

30 
40 

50 



43 

4-3 
5-0 
5-7 
6.4 
7-1 
14-3 

21-5 

28.6 
35-8 



42 

4.2 

4.9 

5-6 
6.4 
7-1 
141 
21 .2 
28.3 
35-4 



31 

3-1 
3.6 
4.1 

4-6 
5-1 



25-8 



30 

30 
3-5 
4.0 

4-5 
5-0 
10. o 
15.0 
20.0 
25.0 



29 

2.9 

3-4 
3-9 
4.4 

4.9 

9-8 
14.7 

^9-6 
24.6 



P. P. 



416 



TABLE V^III.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

40° 47° 



_9_ 
10 
II 

12 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 
23 
24 



25 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



Log. Vers. I 1> Los- Kxsef 



50 

51 

52 
53 
S±_ 

55 
56 
57 
58 
59 



(»0 



48478 
4S508 
48538 
48568 

4859? 



48627 

48657 
48686 

487 1 6 
48746 



4877^ 
48805 

4S835 
48864 
48894 



48923 

48953 
48983 
49012 
49042 



49071 
49101 

49130 
49160 

49189 



49219 
49248 
49278 

49307 
49336 



49366 

49395 
49425 
49454 
49483 



49513 
49542 

49571 
49601 
49630 



49659 
49689 
49718 
49747 
49776 



49806 

49835 
49864 

49893 
49922 



;oi 



D5 



49952 . 
49981 
50010 
50039 I 
50068 I 



50097 
50126 



50185 
50214 



9.50243 



liOff. Vers. 



29 

30 
29 

30 
29 
29 

30 
29 

29 

30 
29 

29 
29 

29 

30 

29 

29 
29 

29 
29 
29 
29 
29 

29 
29 
29 

29 
29 
29 
29 
29 
29 
29 

29 
29 
29 

29 
29 

29 
29 

29 

29 
29 

29 
29 
29 

29 
29 

29 
29 

29 
29 
29 
29 

29 
29 
29 
29 
29 



J> 



9 



64301 
64344 
64387 
64430 

64473 



64515 

64558 
,64601 
64644 
64687 



64729 
,64772 
64815 
64858 
6490 1 



• 64943 
.64986 
,65029 
,65072 
65114 



9- 



65157 
65200 

65243 
65285 

65328 



65371 
.65414 

.65456 
.65499 
65542 



65585 
,65627 
,65670 

.65713 
6575? 



9- 



65798 
65841 
65884 

65926 
65969 



,66012 
.66054 
. 66097 
. 66 1 40 
.66182 



v> 



,66225 
.66268 
.66310 
•66353 
,66396 



66438 
,66481 
66523 
.66566 
, 66609 



,66651 
, 66694 
,66737 
66779 
66822 



9.66864 



43 
42 

43 
43 
42 
43 
43 
42 
43 
42 
43 
43 
42 
43 
42 
43 
42 

43 

42 

43 

42 

43 
42 
43 
42 

43 
42 

43 
42 

43 
42 

43 

42 

42 

43 
42 
43 
42 
42 

43 
42 
42 

43 
42 

42 
43 
42 
42 
43 
42 
42 
42 

43 
42 

42 
42 

43 
42 
42 
42 



liOij. Vers. 



josr. Kxser. /> 



9.50243 
50272 
50301 
50330 
50359 



50388 
50417 
50446 
50475 
50504 



50533 
50562 

50591 

50619 

50648 



50677 
50706 
50735 
50764 
50793 



50821 
50850 
50879 
50908 
50937 



50905 

50994 
023 
0:;2 



9-5 



Lost. 



080 



109 

138 
167 

195 

224 



253 
281 
310 

338 
367 



/> 



396 
424 

453 
481 
510 



539 
567 
596 
624 
653 



681 
710 

738 
767 

795 
823 
852 
886 
909 
937 
90 5 



29 

29 
29 
29 

29 
29 
29 
29 
29 
29 

29 
29 

28 
29 
29 
29 
28 
29 
29 

28 

29 
29 

28 

29 

28 

29 

28 

29 

28 

29 

28 

29 

28 
28 

29 

28 
28 
28 

29 

28 
28 
28 
28 
28 

29 

28 
28 
28 
28 

28 
28 
28 
28 

28 

28 
28 
28 
28 

28 

28 



/> 



l.dir. Kxscc 



/> 



9- 



66864 

66907 
66950 
66992 

67035 
67077 

67120 
67162 
67205 
67248 



,67296 

67333 

•67375 
,67418 

, 67466 



,67503 
67546 

.67588 

,67631 

,67673 



67716 
67758 

67801 

67843 

67886 

,67928 
,67971 
,68013 
68056 

68098 



9- 



68I4I 
68183 

68226 

68268 
683! I 



68353 
68396 

68438 

68481 

68523 



,68566 
,68608 
.68651 

,68693 

.68735 



,68778 
,68826 
,68863 

,68905 

,68948 



68996 

69033 
69075 

69II7 
69160 

69202 

69245 

69287 

69330 
69372 

60414 



42 

43 

42 
42 
42 
42 
42 

43 

42 
42 
42 
42 
42 
42 
42 

43 

42 
42 
42 

,1 -7 

T- 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 

42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 



9- 



/> 







_9 
16 

1 1 
12 

13 

14 



15 
16 

17 
18 

19 



33 
34 



J5 
36 

37 

3 

39 



8 



40 

41 
42 

43 
44 



45 

46 

47 

48 

49_ 

'lO 

5' 
52 
53 
54 



55 
56 
57 
58 
59 
<>0 



V. v. 



20 

40 
50 



6 

7 
8 

9 
10 
20 
30 
40 
50 



43 42 



6 


4 3 


4-'-s 


7 


5-0 


4.9 


9 


5-7 
6.4 


5-6 
6.4 


10 


7-1 


7' 


20 


»4-3 


14.1 


30 
40 


21.5 
28.6 


21.2 
28.3 


50 


35-8 


35-4 



20 

30 
40 
50 



30 



29 

2.9 

3 4 

3-8 

4-3 



6 

7 

8 

9 

lo 

20 

30 
40 
50 



42 



3 





3 


5 


4 





4 


5 


5 





10 





15 





20 





25 






29 

29 

4 
9 
4 
9 



28 



28 

2.8 

3-2 

3-7 

4-a 

4- 

9 

>4 

18. 

23. 



I'. V 



417 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

48° 49° 







10 

II 

12 



Los. Vers . J> 



15 
i6 

17 
i8 

19 



20 

21 
22 
23 

24 



25 

26 
27 
28 
29 



30 

3f 

32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 

43 
±L 

45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



60 



9.51965 

51994 
52022 
52050 
52079 



52107 

52135 
52164 
52192 
52226 



52249 
52277 
52305 
52333 
52362 



52390 
52418 

52446 
52474 
52503 



52531 
52559 
52587 
52615 

52643 



52671 
52699 
5272^ 
52756 
52784 



52812 
52840 



52896 
52924 



52952 
52980 
53008 
53036 
53064 



53092 
53120 
5314? 
53175 
53203 



53231 
53259 
53287 

53315 
53343 



53370 
53398 
53426 
53454 
53482 



53509 
5353? 
53565 

53593 
53620 



9.53648 



28 
28 

28 

28 



28 
28 

28 



28 

28 

28 
28 

28 

28 

28 

28 
28 
28 

28 

28 



28 

28 

28 
28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 

2? 

28 
28 

28 

27 
28 
28 
28 

2f 

28 
28 

2? 

28 

2f 
28 

2? 
28 

27 
28 



Log. Vers.' J> 



Log. Exsec. 



9.69414 
69457 

69499 
69542 
69584 



69625 
69669 
697 1 1 

69753 
69796 



69838 
69881 
69923 

69965 
70008 



70050 
70092 

70135 
70177 
70220 



2> i Log. Vers. 



70262 
70304 
70347 
70389 

70431 



70474 
70516 

70558 
70601 

70643 



70685 
70728 
70770 
70812 
70854 



70897 
70939 
70981 
024 
066 



I08 

151 

193 

235 
278 



320 
362 

404 
447 
489 

531 

573 
616 

658 
706 



743 
785 
82^ 
869 
912 



9-71954 



Log. Exspc. 



42 
42 

42 
42 

42 

42 
42 
42 
42 

42 
42 
42 

42 

42 

42 
42 
42 
42 
42 

42 
42 
42 

42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

^2 
42 
42 
42 
42 
42 



T> 



53648 
53676 
53704 
53731 

53759 



53787 
53814 
53842 
53870 
5389? 



53925 
53952 
53986 
54008 
54035 



54063 
54096 
54118 

54'45 
54173 



1> Los. Exsec. 



54200 
54228 

54255 
54283 

543'6 



54338 
54365 
54393 
54426 
54448 



54475 
54502 
54530 
54557 
54585 



54612 

54639 
54667 
54694 
54721 



54748 
54776 
54803 
54836 
54858 



54885 
54912 

54939 
54967 
54994 



55021 
55048 
55075 
55103 

55130 



55157 
55184 
5521T 

55238 
55265 



55292 



27 
28 
27 
27 
28 
27 

28 

2? 

27 
28 

27 
27 
2? 

27 
27 

27 
27 
2? 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 

27 
27 

27 
2? 
27 
27 
27 
27 
27 

27 
2l 
27 
27 
27 

27 

2? 

27 

2? 
27 
27 

27 
2? 
27 
27 
27 



71954 
71996 
72038 
72081 
72123 



72165 
7220^ 
72250 
72292 
72334 



72376 
72419 

72461 
72503 

72545 



72587 
72630 
72672 
72714 

72756 



72799 
72841 

72883 

72925 
7296^ 



73010 
73052 
73094 
73136 
73"78 



73221 
73263 
73305 
7334? 
73389 



73431 
73474 
73516 

73558 
73606 



73642 
73685 
73727 
73769 
73811 



73853 
73895 
73938 
73980 
74022 



74064 

74106 

74 '48 
74191 

74233 



74275 
7431? 
74359 
7440T 

74444 



74486 



2) 



Vers. D Lour. Exseo 
418 



42 
42 
42 
42 
42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 

42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 

42 
42 
42 

42 
42 
42 
42 
42 
42 



It 



10 

1 1 

12 

13 
14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 

27 

28 

29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 
43 

44 



45 
46 
47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



(JO 



p. P. 



20 

40 
50 



6 

7 

8 

9 
10 
20 

30 
40 

50 



42 

4.2 
4.9 



5- 
6. 

7- 
14- 



28.3 
35-4 



3-3 



4-3 

4-7 

9-5 

14.2 

19.0 

23-7 



9 


4 


10 


4 


20 


9 


30 


13 


40 


18 


50 


22 



21 

2.7 
3-2 
3-6 



P. P. 



42 

4.2 

4.9 
5.6 
6.3 
7.0 
14.0 



28.0 
35-0 



28 28 



2.8 
3-2 

3-7 
4.2 

4-6 

9-3 

14.0 

18.6 

23 3 



27 

2,7 
3-1 
3.6 
4.0 

4.5 
9.0 

13-5 
18.0 
22. <; 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

50° 51" 



10 

1 1 

12 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



26 

27 
28 
29 



30 

31 
32 
33 
34 



35 

36 
37 
38 
39 



40 

41 
42 

43 

44 



45 
46 

47 
48 

49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



Los. Vers. I 7> Lotf. Kxseo 



9.55292 

.55319 
.55347 
.55374 
.55401 



55428 

55455 
55482 

55509 
555^.6 



9.55563 
.55590 
.55617 

.55644 
.55671 



9.55698 
.55725 
.55751 
•55778 
.5580^ 



9.55832 

.55859 
.55886 

•55913 
.55940 



9.55966 

.55993 
. 56020 

• 56047 
. 56074 



9. 56101 
.56127 
.56154 
.56181 
. 56208 



9.56234 
. 56261 
.56288 

•56315 
.56341 



9.56368 

. 56395 
.56421 
. 56448 
.56475 



9.56501 
.56528 

•56554 
.56581 
. 56608 



9.56634 
.56661 
. 56687 
.56714 
.56741 



9.56767 

• 56794 
.56826 

.56847 

•56873 

9 . 56900 

Lotf. Vers. 



27 
2f 

27 
27 

27 
27 
27 
27 
27 

27 
27 
27 

27 
27 

27 

27 

26 
27 
27 

27 
27 

26 

27 
27 

26 

27 

27 

26 

27 
27 

26 

27 

26 

27 

26 

27 

26 

27 

26 
26 

27 

26 
26 

27 

26 
26 
26 

27 

26 
26 
26 
26 
26 

27 

26 
26 
26 
26 
26 
26 

J) 



J> 



■ 74486 

.74528 

74570 

,74612 

74654 



74696 

74739 

74781 
74823 
7486^ 



, 74907 

■ 74949 
.74991 
.75033 
75076 



,75118 
, 75 1 60 
,75202 

75244 
75286 



75328 
75370 
75413 
.75455 

75497 



75539 
.75581 
,75623 
,7566^ 

.7570^ 



75750 
75792 
.75834 
.75876 
,75918 



.75966 
, 76002 
.76044 
.76086 
,76128 



76171 
,76213 
,76255 
,76297 
.76339 



7638T 
,76423 

.76465 
,76507 

.76549 



,76592 
.76634 
, 76676 
.76718 
.76760 



9.76802 

.76844 

.76886 

.76928 
.76976 

9.77012 

liOC Kxsec I I> 



42 
42 

42 
42 

42 
42 
42 
42 
42 

42 
42 

42 
42 
42 

42 
42 

42 
42 
42 

42 
42 

42 
42 
42 

42 
42 
42 

42 
42 

42 
42 

42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 

42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



Lotf. Vers. I 7> 



9. 56900 
56926 

56953 
56979 
57005 



57032 

57058 
57085 
5711T 
57138 



57164 
57196 

57217 

57243 
57269 



57296 

57322 

57348 
57375 
57401 



57427 
57454 
57480 

57506 

57532 



57559 
57585 
57611 

5763^ 
57664 



57690 
57716 

57742 
57768 
57794 



57821 
57847 
57873 
57899 
57925 



57951 
57977 
58003 
58029 
58055 



58082 
58108 

58134 
58160 
58186 



58212 
58238 
58264 
58290 
58316 



58342 
58367 
58393 
58419 
5^445 

58471 

Lotr. Vers. 



26 
26 
26 

26 

26 
26 
26 
26 
26 

26 

26 
26 
26 

26 

26 
26 

26 

26 
26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 
26 

26 

26 
26 

26 

26 
26 
26 

26 

26 
26 
26 
26 
26 

26 

26 
26 
26 
26 

26 
26 
26 
26 
26 

26 

25 

26 
26 
26 
26 

/> 



iOi;. Hxsec.l /> 



9 



77012 

77055 
77097 

77139 
77181 



7722 '^ 
77265 
7730^ 

77349 
77391 



77433 

77475 
77517 
77560 
77602 



77644 
77686 
77728 
77770 
77812 



77854 
77896 

77938 
77986 
78022 



78064 
78107 

78149 
78191 
78233 



78275 
78317 
78359 
7840T 

78443 



78485 
78527 
78569 
7861 1 
78653 



78696 

78738 
78780 
78822 
78864 



78906 
78948 
78996 

79032 
79074 



79H6 
79'58 
79206 
79242 
79285 

79327 
79369 
794 II 
79453 
79495 
79537 



42 
42 
42 
42 

42 
42 
42 
42 

42 

42 

42 

42 
42 
42 

42 

42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 
42 

42 
42 

42 
42 
42 

42 
42 
42 
42 

42 
42 



Lotf. Kxsec. /> 



10 

1 1 

12 

13 
14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

i2_ 
50 

5« 
52 
53 
54 



55 

56 
57 
58 

(iO 



V. V 



30 
40 
50 



40 
50 



30 
40 

50 



42 42 



13.2 

17-6 
22.1 



28.0 
35.0 



27 27 



2.7 


2. 


3.2 


3- 


3-6 


3 


4.1 
4.6 


4- 
4. 


g.i 


9 


13-7 
18.3 


»3- 
18. 


22.9 


22. 



2g 26 



.6 
3.0 
3-4 
3 9 
4-3 



17. 



2S 



7 


3.0 


8 


3.4 


9 


3.8 


10 


4.2 


20 


8.5 


30 


12.7 


40 


17.0 


50 


21..: 



i». r 



419 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

52° 53° 



10 

II 

12 

13 
14 



15 
i6 

17 



19 



20 

21 

22 

23 

24 



25 
26 

27 

28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



Log. Vers. 



40 

41 

42 
43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



60 



58471 
5S49? 
58523 
58549 
58575 



58601 
58626 

58652 

58678 
58704 



58730 

58755 
58781 
5880^ 
58833 



58859 
58884 
58916 

58936 
58962 



58987 
59013 
59039 
59064 
59096 



59116 
5914T 
5916? 
59193 
592 1 8 



59244 
59270 

59295 
59321 
59346 



59372 
5939f 
59423 
59449 
59474 



59500 
59525 
59551 
59576 
59602 



5962^ 
59653 
59678 
59704 
59729 



59754 
59780 

59805 
59831 
59856 



5988T 
59907 
59932 
59958 
59983 



9 . 60008 



Loj;. Vers. 



D Log. Exsec. ! D 



26 

25 
26 
26 

26 

25 
26 
26 

25 
26 

25 
26 
26 

25 
26 

25 
26 

25 
26 

25 

25 
26 

25 
26 

25 
25 
26 

25 
25 

25 
26 

25 
25 
25 

25 
25 
26 

25 

25 

25 

25 
25 
25 
25 

25 
25 
25 

25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 



D 



79537 

79579 
7962T 

79663 
79705 



7974^ 
79789 
79831 
79874 
79916 



79958 
80000 
80042 
80084 
80126 



80168 
80216 
80252 
80294 
80336 



80378 
80426 
80463 
80505 
80547 



80589 
80631 
80673 

80715 
80757 



80799 
8084T 
80883 
80925 
80968 



010 
052 
094 
136 

178 



220 
262 
304 
346 
388 



430 
473 
515 

557 
599 



641 
683 
725 
76^ 
809 



851 
894 
936 
978 
82020 



82062 



42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 

42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 

42 
42 
42 
42 
42 
42 

42 
42 
42 

42 
42 

42 
42 
42 
42 
42 
42 

42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



Log. Kxsec.i 1) 



Log. Vers. 



9 . 600O8 
60034 
60059 
60084 
60IIO 



60135 
60166 
60185 
6021 I 
60236 



60261 
60285 
60312 
60337 
60362 



6038^ 
60412 
60438 
60463 
60488 



60513 

60538 
60563 
60589 
60614 



60639 
60664 
60689 
60714 
60739 



60764 
60789 
60814 
60839 
60864 



60889 
60914 
60939 
60964 
60989 



014 

039 
064 
089 
114 



139 
164 
189 
214 
239 



264 
289 
313 

338 
363 



388 
413 
438 
462 

48^ 



512 



n 



25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 

25 
25 
24 
25 
25 
25 
24 
25 
25 
25 

2% 

25 
24 

25 
25 



I) 



Log. Exsec. D 



82062 
82104 
82146 
82188 
82236 



82272 
82315 

82357 
82399 
82441 



82483 

82525 
8256^ 
82609 
8265T 



82694 
82736 
82778 
82820 
82862 



82904 
82946 
82988 
83031 
83073 



83115 

83157 

83199 
8324T 

83283 



83325 
83368 
83410 
83452 
83494 



83536 

83578 
83626 
83663 
83705 



83747 
83789 
83831 
83873 
83916 



83958 
84000 
84042 
84084 
84126 



84168 
842 II 

84253 
84295 
8433? 



84379 
84422 
84464 
84506 

84548 



84596 



Log. Exsec. 



42 
42 
42 
42 
42 
42 
42 
42 

42 

42 
42 
42 
42 
42 

42 
42 
42 

42 
42 

42 
42 

42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 

42 
42 

42 
42 
42 

42 
42 
42 
42 
42 
42 



I> 



10 

II 

12 

13 
14 



15 
16 

17 



19 



20 

21 

22 

23 
24 



25 
26 

27 

28 

29 



40 

41 
42 

43 
44 



p. P 



20 

40 
50 



20 

30 
40 

50 



42 

4.2 
4.9 

5-6 
6.4 
7-1 
M-i 
21.2 
28.3 
35-4 



7 


3 


8 


3 


9 


3 


10 


4 


20 


8 


30 


13 


40 


17 


50 


21 



26 



25 



16 



20.8 



42 

4.2 

4.9 

5 6 
6.3 
7.0 
14.0 
21 .0 
28.0 
35-0 



2% 

2-5 
3-0 
3-4 
3-8 
4-2 

8-5 
12.7 

17.0 

21.2 



24 

2.4 
2 8 
3-2 
3-7 
4.1 



16. 



P. P. 



420 



TABLE VIII, —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



54 



r>i> 



5 
6 

7 
8 

9 
10 

II 

12 

13 

U 



15 
i6 

17 
i8 

19 



20 

21 
22 
23 

24 

25 

26 

27 
28 
29 



30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44_ 

45 
46 

47 
48 

49 



L()!r. Vers. 



It 



50 

51 

52 
53 
54 



55 
56 

57 
58 
59 



9.6 
.6 
.6 
.6 
.6 



9.0 
.6 
.6 
.6 
.6 



9.6 
.6 
.6 
.6 
.6 



9.6 
.6 
.6 
.6 
.6 



12 



3 

537 
562 

586 
61T 



636 
661 
685 
716 

735 



760 

784 
809 

834 
858 



883 
908 
932 
957 
982 



9.62005 
.62031 
.62055 
.62086 
.62105 



9.62129 
.62154 
.62178 
.62203 

.62227 



9.62252 
.62275 
.62301 
.62325 
.62350 



9.62374 
.62399 
.62423 
.62448 
.62472 



9.62497 
.62521 
.62546 
.62576 
.62594 



(JO 



9.62619 
.62643 
.62668 
.62692 
.627 1 6 



9.62741 
.62765 
.62789 
.62814 
.62838 



9.62862 

.62887 

6291T 

.62935 

.62960 



9.62984 



24 
25 
24 
25 
24 
25 
24 
25 
24 

25 
24 
24 
25 
24 

25 
24 
24 
24 
25 
24 
24 
24 
25 
24 

24 

24- 

24 

24 

24 

24 

24 
24 

24 
24 
24 

24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 



Ii08?. Kxsec. 1> 



84596 
84632 

84675 
,84717 

84759 
8480T 

84843 
84886 

84928 

84970 



85012 
85054 
85097 

85139 

8;i8i 



9.85223 
.85265 
.85308 

•85350 

.85392 



85434 
85476 
,85519 
,85561 
,85603 



85645 
85688 

85730 
,85775 
,85814 



9- 



85857 
85899 
85941 

85983 
86026 



Lost. Vers. 



/> 



,86068 
86116 
,86152 
.86195 
.86237 



86279 
8632T 
86364 
86406 
86448 



86496 

86533 

86575 

,86617 

86659 



86702 

86744 

86786 

.86829 

,86871 



86913 
86956 
86998 
87046 
87082 



9.87125 



Los;. Kxspr. 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 

42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 

42 
42 
42 



Lotf. \i'\<. 



It I. 



9.62984 
•630O8 
.63032 
.63057 
.63081 



9.63105 
.63129 

•63154 
.63178 
.63202 



9.63226 
.63256 

.63274 
.63299 

■63323 



9-63347 
•63371 
•63393 
•63419 
•63443 



9.63468 
.63492 
.63516 
•63540 
•63564 



9.63588 
.63612 
•63636 
. 63666 
•63684 

9-63708 
•63732 

.63756 
•63786 
•63804 



9.63828 
.63852 
.63876 
.63900 
.63924 



7> 



9.63948 
.63972 
.63996 
.64019 
• 64043 



9.64067 
.64091 
.64115 

.64139 
.64163 



9.64187 
.64216 
.64234 

.64258 
.64282 



9.64306 
.64330 
.64353 
.64377 
. 6440 1 



9.64425 



24 

24 
24 
24 

24 
24 

24 

24 
24 

24 
24 

24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 

24 
24 

24 

24 

24 
24 
24 
24 
24 

24 

24 
24 
24 
24 
24 
24 
24 

23 
24 

24 
24 
24 

23 
24 

24 

23 
24 
24 
23 
24 
24 
-J 
24 
23 
24 



I,Off. Vers. /> I,oc. Kxser 



Kxser 



n 



CS7125 
87167 
87209 
87252 
87294 



87336 
87379 
87421 

87463 
87506 



87548 
87596 

87633 
87675 
87717 



87760 
87802 
87844 
87887 
87929 



87971 
88014 
88056 
88099 
88141 



88183 
88226 
88268 
S8316 

88353 



88395 
88438 
88486 
88522 
88565 



88607 
88650 
88692 
88734 
88777 



88819 
88862 
88904 

88947 
88989 



8903 T 

89074 
89116 
89159 
8920T 

89244 
89286 
89329 

8937T 
89414 



89456 
89499 
89541 
89583 
89626 

89668 



42 
42 
42 
42 

42 
42 

42 
42 

42 

42 
42 
42 
42 
42 

42 

42 
42 
42 

42 

42 
42 
42 

42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



_4_ 

5 
6 

7 
8 

9 
10 

1 1 

12 

13 
14 



15 
16 

17 

18 

19 



•20 

21 

->2 

23 

24 



-5 
26 
27 
28 
29 



30 



32 

1 '^ 

34 



35 
36 

37 

3 

39 



8 



40 

41 
42 
43 
44 



i> 



45 
46 
47 
48 
49 

:>() 

5' 
52 
53 
54 



55 
56 
57 
58 
59 
<>0 



I'. I'. 



20 

30 
40 

50 



40 



20 

30 
40 
50 



42 

42 
4.9 

5-6 
6.4 
7 » 

14. 1 

21 .2 
28.3 
35.4 



24 



16 



42 

4 a 

4.9 

56 

6.3 

7.0 

14.0 

21 .0 

28.0 

35.0 



25 


24 


2 5 


2 


2 


9 


2. 


3 


3 


3. 


3 


7 


3. 


4 


I 


4 


8 


3 


8 


12 


S 


12 


16 


6 


16. 


20 


8 


20 



23 



3.» 
3-5 
3-9 
7-8 
II. 7 

15-6 
19.6 



r. I' 



421 



TABLE VIIL— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

56° 57 



10 

II 

12 

14 



15 
16 

18 
19 



20 

21 

22 
23 
24 



25 
26 

27 
28 
29 



30 

31 

32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 

43 
44 



45 

46 

47 
48 

49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
00 



Log. Vers. 



9.64425 
64448 
64472 
64496 
64520 



64543 
6456^ 
64591 
64614 
64638 



64662 
6468I 
64709 

64733 
64756 



64786 
64804 
6482? 
64851 
64875 



64898 
64922 

64945 
64969 
64992 



D Lost. Exsec. D 



650I6 
65040 
65063 
65087 
651 16 



65134 
65157 
65181 
65204 
65228 



65251 
65275 
65298 
6532T 

65345 



65368 
65392 
65415 

65439 
65462 



65485 
65509 
65532 
65556 
65579 



65602 
65626 

65649 
65672 
65696 



65719 
65742 

65765 
65789 
65812 



65835 



Log. Vers. 



23 
24 
23 

24 

23 

24 
23 
23 
24 

23 
23 
24 
23 
23 
24 
23 
23 
23 
24 

23 
23 
23 
23 
23 
24 
23 
23 
23 
23 

23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 

23 
23 
23 
23 

23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 
23 



89668 
897 1 1 

89753 
89796 
89838 



89881 

89923 
89966 

90OO8 
90051 



90094 

90136 
90179 
90221 
90264 



903O6 
90349 
90391 

90434 
90476 



90519 
90561 
90604 

90647 
90689 



90732 

90774 
90817 
90860 
90902 



90945 
90987 



030 
073 



158 
200 

243 
286 

328 



371 
414 

456 
499 
541 



584 
627 
669 
712 

755 



79? 
846 

883 

926 

968 



92011 
92054 
92096 
92139 
92182 



92224 



42 
42 

42 
42 

42 
42 
42 
42 

42 

43 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 

43 

42 

42 

42 
42 

42 

43 

42 

42 
42 
42 
43 

42 

42 
42 
42 
43 
42 
42 

43 
42 
42 
42 

43 
42 
42 
43 
42 

42 

43 
42 

43 
42 

42 

43 

42 

43 

42 

42 



D Log. Exsec. !> 



Log. Vers. 



9.65835 

.65859 
.65882 

.65905 

.65928 



9.65952 

.65975 
.65998 
.66021 
. 66044 



9.66068 
.66091 
.66114 

•66i3f 
. 66 I 66 



9.66183 
.66207 
.66230 
•66253 
.66276 



9.66299 
.66322 

.66345 
.66368 
.6639I 



9.66415 
.66438 
. 6646 I 
.66484 
.66507 



D Log. Exsec. Z> 



9.66530 
•66553 
.66576 
•66599 
.66622 



9.66645 

.66668 
.66691 
.66714 

■66737 



9.66760 
.66783 
.66805 
.66828 
.66851 



9.66874 
.6689^ 
. 66926 
.66943 
.66966 



9.66989 
.67012 
.67034 
.6705; 
.67086 



9.67103 
.67126 
.67149 
.67171 
.67194 



9.67217 



23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 
23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 
23 
23 

23 

22 

23 

23 
23 

22 

23 

23 
22 
23 
23 

22 

23 

22 



Loe. Vers. 



I) 



92224 
92267 
92310 
92353 
92395 



92438 
92481 
92524 

92566 
92609 



92052 
92695 

9273? 
92786 

92823 



92866 
92909 
92951 
92994 
93037 



93080 
93123 
93165 
932O8 

9315J 
93294 
93337 
93380 
93422 
93465 



93508 
93551 
93594 
93637 
93680 



93722 
93765 
93808 
93851 
93894 



93937 
93980 
94023 
94066 
94109 



941 51 
94194 

94237 
94286 

94323 



94306 
94409 

94452 
94495 
94538 



9458T 
94624 
9466^ 
94716 

94753 



94796 



Log. Exsec. T> 



43 

42 
43 
42 

43 
42 
43 
42 
43 
42 
43 
42 
43 
42 

43 

43 
42 
43 
42 

43 
43 
42 
43 
43 
42 
43 
43 
42 
43 

43 
42 
43 
43 
43 
42 

43 
43 

43 
42 

43 
43 
43 
43 
43 
42 
43 
43 
43 
43 

43 
43 
43 
43 

43 

43 
43 
43 
43 
43 
43 



5 
6 

7 
8 

_9_ 
10 

II 
12 

13 

14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 
27 
28 
29 



30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 

43 

44 



45 
46 
47 
48 
49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



60 



p. P. 



40 
50 



20 

40 
50 



40 
50 



43 



4 


3 


5 





S 


7 


6 


4 


7 


I 


M 


3 


21 


5 


28 


6 


35 


8 



24 

2-4 
2.8 

3.2 

3-6 
4 o 



16. 



42 

4.2 

4.9 

5-6 
6.4 
7-1 

14. 1 

21 .2 
28.3 
35-4 



23 



3.1 
3-5 
3.9 

7-§ 
11.7 

19.6 



23 


2 


2.3 


2. 


2.7 


2. 


3.0 


3 


3-4 


3- 


3-8 


3. 


7^6 


7- 


II-5 


II. 


15-3 


T.S- 


19.1 


18. 



P. p. 



422 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

58° 59" 



Lo:;. Vers. 



1* 







9 
10 

1 1 

12 



15 
i6 

i8 
19 



20 

21 

22 

23 

24 



-5 
26 
27 
28 

-9_ 
30 

31 
32 
33 
34 



35 
36 
37 
38 

39 



40 

41 

42 
43 
44 

45 
46 
47 
48 

49 



50 

51 
52 
53 
54 

55 
56 
57 
5S 
59 
00 



9.67217 
.67240 
.67263 
.67285 
•67308 



9.67331 
.67354 
•67376 
.67399 
.67422 



9.67445 
.67467 
.67490 

.67513 
•67535* 



9.67671 
.67694 
.67717 

•67739 

.67762 



9.67784 
.67807 
•67830 
.67852 
•67875 



9.67897 
.67920 
.67942 
.67965 
.67987 



9.68010 
.68032 
.68055 
.68077 
.68 1 00 



9.67558 
.67581 
.67603 
.67626 
.67649 



9.68122 
.68145 
.68167 
. 68 1 90 
.68212 



9.68235 
.68257 
.68280 
.68302 
•68324 



9.68347 
•68369 
.68392 
.68414 
•68436 



9.68459 

.68481 

•68503 

.68526 

•68548 
9-68571 

Loir. Vers. 1 /> 



Loir. Kxsec. 



/> 



-^3 

-3 
22 

23 

22 

23 
22 

23 






22 

23 
22 

22 

22 
22 
23 



22 



22 
22 
22 
22 

22 
22 

-IT 



22 



9^94796 

94839 
94882 

94925 
94968 



95011 

95054 
95097 
95140 
95183 



95226 
95269 

95313 
95356 
95_399 
95442 

95485 
95528 

95571 
95614 



95657 
95700 

95744 
95787 
95830 



95873 

959'6 

95959 
96002 

96046 



96089 
96132 
96175 

962 1 8 
9626T 

96305 
96348 

96391 
96434 
96478 



96521 
96564 
96607 
96656 
96694 



96737 
96786 
96824 
96867 
969 1 6 

96953 
96997 
97040 
97083 
97127 



97170 
97213 
97257 
97300 

97343 
9^97387 

IjOjf. Kxser. 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 

43 

43 
43 
43 
43 

~~Tr 



\A)\i. \('IS. 



It 



68571 
.68593 
,68615 
,68637 
,68660 



,68682 
,68704 

,68727 
,68749 
,68771 



68793 
,68816 
,68838 

,68866 
,68882 



68905 
,68927 
, 68949 
,68971 
,68993 



,69016 
.69038 
. 69060 
.69082 
.69104 



69126 

69149 
691 7 1 
,69193 
,6921 5 



69237 
.69259 
,69281 
69303 
,69325 



69347 
,69369 
,69392 
.69414 
,69436 



,69458 
,69480 
.69502 
,69524 
,69546 



,69568 

69590 
,69612 
,69634 
,69656 



69678 
69700 
,69721 

69743 
69765 



9.6978^ 
. 69809 
.69831 
.69853 
•69875 
9 . 69897 
liOsr. V<'rs. 



22 
22 
22 

22 
22 

22 

22 
22 

22 
22 
22 
22 

22 

22 
22 
22 
22 
22 

22 

22 

22 
22 
22 

22 
22 
22 
22 

22 

22 
22 
22 
22 
22 

22 
22 

22 

2 2 

O -> 

22 
22 
22 

22 
22 
22 

22 
22 

22 
22 
21 
22 
2 2 

2 2 
^2 

21 



l-o:: 



9 
10 



K\MM' 



/> 



97387 
97430 

97473 
97517 
97566 



97603 

97647 
97696 

97734 
97777 



97826 
97864 
97907 
97951 
97994 



98038 
9808 T 
98125 
98168 
982 1 T 



98255 

98298 
983421 
983851 
98429 



98472 
98516 

98559 
98603 
98647 



98696 
98734 
98777 
98821 
98864 

98908 
98952 
98995 

99039 
99082 



99126 
99170 
99213 
99257 
99300 



99344 
99388 

99431 
99475 
995 '9 
99562 
99606 
99650 
99694 
99737 



99781 
99825! 
99868! 
99912 

99956 
00000 I 

KxsHr. 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
44 
43 

43 
43 
43 
43 
43 

43 
44 
43 
43 
43 

43 
44 
43 
43 
43 
44 
43 
43 
44 
43 

43 
44 
43 
44 
43 
44 
43 
43 
44 
43 
44 

"TT 



{) 



4 

5 
6 

7 
8 

_9 
10 

1 1 
12 

13 
14 

15 
16 

17 
18 

19 



20 

21 

22 

23 
24_ 

25 
26 

27 
28 

30 

31 

32 

35 
36 

37 
38 
39 



40 

4' 
42 
43 
44 

45 
46 

47 
48 
49 

:a) 

51 
52 

53 

5-+_ 

55 

56 

57 

58 

59 

(>0 



v. V. 



20 
30 
40 

so 



40 
50 



30 
40 

50 



44 



4.4 


4- 


5-1 


5 


5.8 
6.6 


5 
6 


7.3 


7 


M.6 


14. 


22. 


21 


29.3 
36.6 


29. 
36 



43 



43 



6 


4 3 


7 


5.0 


8 


5 7 


9 


6 4 


10 


7-1- 


20 


14 3 


30 


21-5 


40 


28.6 


50 


35. 8 



23 

2.3 

2.7 

3^o 



19. 



22 

2.2 
2.6 



15.0 
18.7 



22 



21 



32 
3-6 

7-1 
10.7 

14.3 

17.9 



423 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

60° 61° 



10 

II 

12 

13 

14 



15 

i6 

17 
i8 

19 



20 

21 

22 

23 

24 



26 
27 



29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



60 



Log. Vers, 



9.69897 
.69919 
. 69946 
. 69962 
.69984 



9 -70005 
. 70028 
. 70050 
.70072 
• 70093 



9-70115 
.70137 
.70159 
.70181 
.70202 



D 



9.70224 
. 70246 
.70268 
.70289 
.70311 



9-70333 

•70355 

.70376 

. 70398 
.70420 



9.70441 
.70463 
.70485 
.70507 
•70528 



9.70550 
.70572 

• 70593 
.70615 

• 70636 



9.70658 
. 70680 
.70701 
.70723 

.70745 



9.70765 
.70788 
. 70809 
.70831 
.70852 



9.70874 
.70896 
.7091^ 

.70939 
, 70966 



9.70982 
.71003 
.71025 
.71045 
.71068 



9.71089 
.71III 
.71132 

.71154 
.71175 



9.71197 



Log. Vers. 



22 
21 

22 
22 

22 
21 
22 
22 
21 

22 
21 

22 
22- 
21 

22 
21 

22 
21 
22 

21 

22 
21 
22 
21 

21 
22 
21 
22 
21 

21 
22 
21 
21 
21 

22 
21 
21 
21 
22 

21 
21 
21 

21 
21 

22 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 
21 



Log. F.xsec. 



10.00000 
. 00044 
.0008^ 
.00131 
. 00 1 7 5 



I o . 002 1 9 
.00262 
.00305 
.00356 
.00394 



10.00438 
.00482 
.00525 
.00569 
. 006 1 3 



D Log. Vers. D 



10.00657 
.00701 
.00745 
.00789 
.00833 



10.00875 
. 00926 
. 00964 

.oioog 
.01052 



10.01 095 
.01 146 
.oi 184 

.OI228 
.01272 



IO.OI315 
.01366 
.01404 

•OI448 
.01492 



10.01535 
.01586 
.01624 
.01668 
.01712 



10.01755 
.01806 
.01844 
.01889 

.01933 



10.01977 
.02021 
.02065 
.02109 
.02153 



10.02197 
.02242 
.02286 
.02330 

.02374 



1 0.024 1 8 
.02463 
.02507 
.02551 
.02595 



10.02639 



J> Log. Exsec. 



44 
43 
44 
43 
44 
. 43 
44 
44 
43 

44 
44 
43 
44 
44 
44 
43 
44 
44 
44 

43 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 



197 
218 

239 
261 
282 



304 
325 
346 
368 

389 



411 
432 
453 

475 
496 



51? 
539 
566 
581 
603 



624 

645 
667 
688 
709 



730 

752 

773 
794 
815 



^1>7 
858 

879 
906 

922 

71943 
71964 

71985 
72005 
72028 



72049 
72070 
7209T 
721 12 
72133 



72154 
72176 
72197 
72218 
72239 



72266 
72281 
72302 
72323 
72344 



72365 

72386 
72408 
72429 
72450 



72471 



I) I Loff. Vers. 



21 
21 
21 
21 
21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 

21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 

21 
21 
21 
21 

21 

21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 



Log. Exsec. 



10.02639 
.02684 
.02728 
.02772 
.02815 



10.02861 
.02905 
.02949 
.02994 
.03038 



10.03082 
.03127 
.03171 
.03215 
.03260 



10.03304 

•03348 
.03393 
•0343^ 
.03481 



10.03526 
.03576 
.03615 
.03659 
.03704 



I0.03748 

-03793 
.03837 

.03881 

.03926 



n 



10.03970 
. 040 1 5 
.04059 
.04104 
.04149 



10.04193 
.04238 
.04282 
.04327 
-04371 



10.04416 
. 0446 I 
.04505 
.04550 
-04594 



10.04639 
. 04684 
.04728 

.04773 
.04818 



10.04862 
.0490^ 
.04952 
.04995 
.05041 



10.05086 
.05131 
.05175 
,05226 
.05265 



10.05310 



I) 



44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
45 
44 
44 
44 
44 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 
45 
44 
45 
44 
44 
45 
44 
45 
44 
45 
44 
45 



10 



20 

21 

22 

23 

24 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 
49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



Log. Exsec. 1> 



GO 



40 
50 



40 
50 



p. P. 



45 

4-5 
5-2 
6.0 
6.7 
7-5 



20 


T5.0 


30 


22.5 


40 


30.0 


50 


37-5 



44 

4.4 

5-i 
5-8 
6.6 

7-3 
14-6 



29-3 
36.6 



44 

4.4 

5-2 

5.9 

6.7 

7-4 
M-8 
22.2 

29-6 
37.1 



43 

4-3 
5-1 
5-8 
6.5 
7.2 

14.5 
21.7 
:>g.o 
36.2 



22 


21 


2.2 


2. 


2-5 


2. 


2.9 


2 


3-3 


3- 


,3-6 


3 . 


^7-3 


7- 


II. 


10. 


14.6 


14. 


18.3 


17- 



^.6 
I 
7 
3 
9 



21 



20 

30 
40 

50 



P. P. 



424 



TABLE VIII.-L0GARITHM;C VERSED SINES AND EXTERNAL SECANTS 

63° 63° 



10 

II 

12 

13 

14 



Lop. Vers. I 7> 



9.72471 
.72492 

•72513 
•72534 
.72555 



9.72576 
.72597 
.72618 
.72639 
.72660 



15 
16 

17 
18 

19 



9.72681 
.72701 

.72722 

.72743 
.72764 



9.72785 
.72806 
.72S2J 
.72848 
.72869 



9.72890 
.7291 I 
.72931 
.72952 
.72973 



9.72994 

.73015 
•73036 
.73057 
•73077 



9.73098 
•73119 
.73140 
.73161 
.73181 



9.73202 
.73223 

•73244 
•73265 
•73285 



9-73306 
•73327 
.73348 

.73368 
•73389 



9-734IO 

.73430 

•73451 
•73472 

•73493 



9-73513 
•73534 
•73555 
.73575 
•73596 



,73617 

.7363^ 
73(>S^ 

73679 
73699 



9.73720 



Loer. Vers. 



21 
21 
21 
21 

21 
21 
21 
21 
21 

2r 

20 

21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
20 
21 
21 

21 

20 

21 

21 

20 

21 

21 

20 

21 

26 

21 

21 

20 

21 

20 

21 
26 
21 
20 
21 

20 

20 

21 

20 

21 

20 

20 

21 

20 

26 

21 

20 

26 

21 

20 

26 



/> 



Loir. Kxsec' J> 



10.05310 
•05354 
•05399 
•05444 
.05489 



10.05534 

.05579 
•05623 

.03668 

.05713 



I0.05758 
.05803 
.05848 
.05893 
.05938 



10.05983 
.06028 
.06072 
. 06 1 1 ^ 
.06162 

10.06207J 
.0625^' ^^ 



44 
45 
45 
44 

45 
45 
44 
45 
45 

45 
44 
45 
45 

45 

45 
45 
44 
45 
45 



.06297 
.06342 
.0638^ 



10.06432 
.0647^ 
.06522 
.06568 
.06613 



10.06658 
.06703 
.06748 
•06793 
.06838 



10.06883 
.06928 
.06974 
.07019 
. 07064 



10,07109 

.07154 
.07200 

.07245 
.07290 



10.0733$ 
•07380 
.07426 
.07471 

.07516 



10.07562 
.07607 
.07652 
.0769^ 
•07743 



10.07788 
.07834 

.07879 
•07924 
.07970 



10.08015 



iOir. Kxs«'c.l 



45 
45 
45 

45 
45 
45 
45 

45 

45 
45 

45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

— 



Loj?. Vers. I 1* 



9.73720 
.73740 
.73761 
•73782 
.73802 



9.73823 

.73843 
.73864 

•7?>^H 
.73905 



9.73926 
.73946 
.73967 
.73987 
. 74008 



).74028 

. 74049 
. 74069 

. 74090 
.74110 



>.74i3i 

.74151 
.74172 

.74192 
.74213 



9.74233 
.74254 
.74274 
.74294 

.74315 



9-74335 
•74356 
•75376 
•74396 
.74417 



9-74437 
.74458 
.74478 
. 74498 
•74519 



9^74539 
•74559 
.74580 
. 74606 
.74626 



9.74641 
.74661 
.74681 
.74702 
.74722 



9.74742 
•74762 

•74783 
•74803 
•74823 



9.74844 
.74864 

•748S4 
.74904 
•74924 



20 
I 26 

21 
I 26 
, 26 

I ~° 
26 

26 

21 

20 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

20 
26 
20 
26 
26 
26 
26 
26 
20 
26 

26 
26 
20 
26 
26 

20 
20 
26 
26 
20 
26 
26 
20 
26 
20 

26 
20 
26 
26 
20 

26 
20 
26 
20 
20 
26 



Lop. Kxsec. 



10.0801 5 
. 0806 I 
. 08 1 06 
.08151 
.08197 



/> 



10.08242 
.0S288 
•08333 
■08379 
,08424 



10.08470 

•08515 
,08561 
.08605 
.08652 



45 
45 
45 

45 

45 
45 
45 



10.08697 

.08743 
,08789 
.08834 
.08880 



10.08926 
.08971 
.09017 
,09062 

-09I08 



10.09154 
. 09200 

.09245 

,09291 

-09337 



10.09382 

.09428 
•09474 

,09520 
.09566 



I0.096IT 

•0965? 

• 09703 

• 09749 
.09795 



10.09841 

.09886! 

.09932! 

■0997 8: 
. 10024 



10. 10070 
.10116 
. 10162', 

. I0208| 

, 10254 



10. 10300 

.10346 
.10392 
.10438 

. 10484 



10. 10530 
•I0576 
. 10622 
. 10668 
,10714 



9.74945 ' 10. 10766 

Loi.'.Vors. /> ILocr. Kxsfr.l 
425 



45 

45 
45 

45 

45 
45 
45 

45 

46 

45 
45 
45 

46 

45 
45 
45 

46 

45 

46 

45 
45 

46 

45 

46 
46 

45 

46 

45 

46 
46 

45 

46 
46 

45 

46 
46 
46 
46 
46 
46 

45 

46 
46 
46 
46 
46 

46 

46 
46 
46 
46 
46 
46 

/> 



5 
6 

7 
8 

_9 

10 

1 1 

12 

13 
14 

15 
16 

17 
18 

19 

20 

21 

22 

23 
24 



I'. I'. 



25 
26 

27 

28 

29 



30 

31 
32 
33 
34 



J3 
36 

37 
38 
39 



52 



55 
56 
57 
58 
59 



(>0 



20 

40 

5^ 



30 
40 

50 



20 

30 
40 

50 



48 46 



5-4 

6 2 

7 o 
7-7 

«5-S 
23.2 
31.0 
38.7 



6 f 
6.9 

7 h 
«5-3 
23.0 

30 6 
38 3 



45 

4-5 
5-3 
6 o 

6 8 

7 6 
15 I 
22.7 

30-3 
37-9 



45 

4-.S 

5 2 
6.0 

6.7 

7-5 

150 

22 5 
30.0 

37 5 



30 
40 

50 



44 

4-4 

5 2 

5 9 

6 7 

7 4 

'4. a 
22.2 

29-6 
37 1 



21 



2 4 

2 8 

31 

3 

10. 
ij, 
'7 



20 





20 


6 


2,0 


7 
8 


'•3 

2-6 


9 


3^o 


10 

30 


3-3 
6-6 


30 


10.0 


40 


•3 3 


50 


16.6 



I', r. 



TABLE VIII. — LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



64 



65 



10 

II 

12 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 
23 
24 



25 
26 
27 
28 
29 



30 

31 
32 
33 

34 



35 
36 
37 
38 
39 



40 

41 
42 

43 
44 



45 

46 

47 
48 

49 



60 

51 

52 
53 
54 



Log. Vers. 



55 
56 
57 
58 
59 
60 



9-74945 

74965 

74985 
75005 

75026 



75046 
75066 
75086 
75106 
75126 



75147 
75167 
75187 
75207 

7522^ 



n Los 



7524? 
7526^ 
75287 
75308 
75328 



75348 
75368 
75388 
75408 
75428 



75448 
75468 
75488 
75508 
75528 



75548 
75568 
75588 
75608 

75628 



75648 
75668 
75688 
75708 
75728 



75748 
75768 
75788 
75808 
75828 



75848 
75868 
75888 
75908 
75928 



75947 
7596^ 

75987 
76007 
76027 



76047 
76067 
76087 
76 log 
76126 

76146 
Log. Vers. 



20 
26 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
26 
20 

20 
20 
20 
20 
26 

20 
20 
20 
20 
20 

20 
20 

20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 

19 
20 
20 
20 
20 
20 

19 
20 

20 

20 

19 
20 
20 
20 

19 
20 

20 
Z> 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 
Log 



Exsec. 



0766 
0807 

0853 
0899 

0945 



0991 

i03f 
1084 
1 1 30 
IJ76 



1222 
1269 

1315 
1 361 

1407 



1454 
1506 

1546 

1593 
1639 



1685 
1732 

1778 
1825 
187T 



JD 



I Log. Vers. 



191^ 
1964 
2010 
2057 
2103 



2150 

2196 

2243 
2289 

2336 



2383 
2429 
2476 
2522 
2569 



2616 
2662 
2709 
2756 
2802 



2849 
2896 
2942 
2989 
3036 



3083 
3130 
3170 
3223 
3270 



3317 
3364 
341 1 

345^ 
3504 

3551 

Exsec. 



46 
46 
46 

46 
46 
46 

46 
46 

46 
46 

46 
46 

46 

46 

46 
46 
46 
46 
46 
46 
46 
46 
4g 
46 
46 
46 
46 
46 
46 

46 
46 
46 
46 
46 
47 
46 
46 
46 
46 
47 
46 
46 
47 
46 

46 
47 
46 
47 
46 

47 
47 
46 
47 
46 
47 
47 
47 
46 
47 
47 



9.76146 
.76166 
.76186 
.76206 
.76225 



9.76245 
.76265 
.76285 

• 76304 

.76324 



9.76344 
. 76364 
.76384 
.76403 

.76423 



9.76443 
.76463 
.76482 
.76502 
.76522 



9.76541 
.76561 
.76581 
. 76606 
.76626 



9 . 76640 
.76659 

.76679 
.76699 

.76718 



X) 



9.76738 
.76758 
.7677^ 
.76797 
.76817 



9.76836 
.76856 

.76875 
.76895 
.76915 



9.76934 
.76954 
.76973 

. 76993 
.77012 



9.77032 
.77052 
.77071 
.77091 
.77116 



9.77130 

.77149 
.77169 

.77188 
.77208 



9.7722^ 
.77247 
.77266 
.77286 

•77305 
9.77325 

Log. Vers. 



19 
20 
20 

19 
20 

19 
20 

19 

20 

20 

19 
20 

19 
20 

19 
20 

19 

19 
20 

19 
20 

19 

19 
20 

19 

19 
20 

19 
19 
20 

19 

19 

19 
20 

19 

19 
19 
20 

19 

19 
19 
19 
19 
19 
20 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 
19 



Log 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 
Log 



Exsec. 



3551 

3598 
3645 
3692 

3739 
3786 

3833 
3886 

392? 
3974 



D 



4021 
4068 
4115 
4162 
4210 



4257 
4304 
4355 

4398 
4445 



4493 
4540 

4587 
4634 
4682 



4729 

4776 
4823 

4871 

491 8 



4965 
5013 
5066 
5108 

5155 



5202 
5250 
529? 
5345 
5392 



5440 
548^ 

5535 
5582 
5630 



5678 
5725 

5773 
5826 

5868 



5916 

5963 
601 1 

6059 
6106 



6154 
6202 
6250 
6298 
6345 
6393 



47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
Al 
47 
47 
47 
4f 
47 

Al 
A7 
47 
4? 
Al 

A7 
Al 
47 
4? 
Al 
47 
Al 
Al 
Al 
47 
Al 
Al 
Al 
Al 
Al 
Al 
Al 
Al 
Al 
Al 
48 
Al 
Al 
Al 
48 

Al 
Al 
48 

47 
Al 
48 
Al 
48 
48 
Al 
48 

IT 



20 



p. p. 



25 
26 

27 
28 

29 



30 

31 

32 
33 

34 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 



45 
46 
47 
48 
49 



50 

51 

52 
53 
54 




20 

40 
50 



40 
50 



20 

30 
40 

50 



48 

4.8 

5-6 

6.4 

7.2 

8.0 

16.0 

24.0 

32.0 

40.0 



47 

4-7 

5.5 

6.2 

7.0 

7.8 

15-6 

23.5 

3i§ 

39.1 



20 



47 

4-7 
5-5 
6.3 
7.1 
7-9 

15. a 

23 -7 
31-6 
39-6 



46 

4.6 

5.4 

6.2 

7.0 

7.7 

15-5 

23.2 

31.0 

38.7 



46 



6 


4.6 


7 
8 


5-3 
6,1 


9 


6.9 


10 


7-6 


20 


15-3 


30 


23.0 


40 
50 


30-6 
38.3 



20 

2.0 

2-3 
2-6 

3.0 
3.3 

6.6 
10. o 

'3-3 
16.6 



19 



6 


1 


9 


7 
8 


2 

2 


3 
6 


9 


2 


9 


10 


3 


2 


20 


6 


5 


30 


9 


7 


40 
50 


13 
16 



2 



P. p. 



426 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



G6 



07° 



Los. Vers. 



10 

II 

12 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 

24 



-3 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 

42 

43 

44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 
60 



77325 
77344 
77363 

77383 
77402 



77422 

77441 
77461 
77480 
774Q9 



775'9 
77538 
7755? 

77 S9G 



77616 

77635 
77654 
77674 
77693 



77712 
77732 

77751 
77770 
77790 



77809 

77828 

77847 
77867 

77886 



77905 

77925 
77944 

77963 

77982 



78002 
78021 
78040 
78059 
78078 



78098 

78117 

78136 

78155 
78174 



78194 
78213 
78232 
78251 
78276 



78289 
78309 
78328 
78347 
78366 



78385 
78404 

78423 
78442 
78462 

9-78481 
Loe. Vers, 



J> 



LoK 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



1) I -OK 



10 



10 



xsec. 7> 



6393 

644 T 
6489 

6537 
6585 



6633 
6680 
6728 

6776 
6824 



6872 
6926 
6968 

7016 
7064 



71 12 
7 1 661 
7209* 

72571 
7305' 



7353 
7401 

7449 
7498, 
7546I 



7594 
7642 
7696 

7739 
778? 



7835 
7884 
7932 
7986 
8029 



807? 
8126 

8174 
8222 
8271 



8319 
8368 
8416 
8465 
8514 



8562 
861 1 
86:^9 
8708 
8757 



8805 

8854 
8903 
895T 
9006 



9049 
9098 
9146 
91951 
92441 

9293! 



48 

47 
48 

48 

48 

4? 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 

48 
48 
48 

48 

48 
48 

48 
48 

48 
48 
48 
48 
48 
48 
48 
48 
48 
48 

48 
48 
48 
48 
48 
48 
48 
48 
49 
48 

48 
48 
48 
48 
49 

48 
48 
49 
48 
49 

48 
49 
48 
49 
49 
48 



liOjf. Vers. 



9.78481 
78500 
78519 
78538 
78557 



78576 

78595 
78614 

78633 
78652 



78671 
78696 
78709 
78728 
7874? 



78766 
78785 
78804 
78823 
78842 



78861 
78886 
78899 

789I8 
78937 



78956 
78975 
78994 
79013 
79032 



79051 
79069 
79088 
79107 

79126 



79145 
79164 

79183 
79202 
79226 



79239 

79258 
79277 
79296 
79315 



79333 
79352 
79371 
79390 
79409 



7942? 
79446 
79465 
79484 
79503 



79521 
79540 
79559 
79578 
79596 
0.79615 

7> \ liOi:. Vers. 



/> \avj: 



9 
9 
9 
9 

9 
9 
9 
9 
9 

9 
9 
9 
9 
9 

9 
9 
9 
9 
9 

9 
9 
9 
9 
8 

9 
9 
9 
9 
9 

9 
8 
9 
9 
9 

9 
8 
9 
9 
8 

9 
9 
8 
9 
9 

8 
9 
9 
8 
9 

8 
9 
9 
8 
9 

8 
9 
8 
9 
8 
9 

TT 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



K\scc 



•9293 
19342 

1 939 1 
19439 
19488 



19537 

19586 

19635 
19684 

19733 



19782 
1983T 
19886 
19929 
19979 



20028 
20077 
20126 
20175 
20224 



20273 
20323 
20372 
2042! 
20476 



20520 
20569 
206 18 
20668 
2071^ 



20767 
20816 
20865 
20915 
20964 



014 
063 

113 
162 
212 



262 

3" 

361 
416 
466 



510 

560 
609 
659 
709 



759 
808 

858 

908 
958 



22008 
22058 
22108 

221581 
22208 

22258I 

Kxser. 



/> 



49 
49 
48 
49 
49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
50 

49 
49 
49 
49 
50 

49 
50 
49 
50 
49 
50 
49 
50 
50 
49 

50 
50 
50 
50 
50 
50 

It 



10 

I 
2 

3 
4 

5 
6 

7 
8 



40 

41 
42 
43 
44 

45 
46 

47 
48 

±9 
oO 

51 
52 
53 

55 
^6 

57 
58 
59 

<;o 



20 
30 
40 
50 



6 

7 
8 

q 
10 
20 
30 
40 
50 



20 

30 
40 
50 



I'. 1* 



49 



4.6 

5-6 

6.4 

7.2 

8.0 

16 o 

24 o 

32 o 

40.0 





19 


6 


1.9 


7 


2-3 


8 


2.6 


9 


2.9 


10 


3.2 


20 


6.5 


30 


9-7 


40 


13.0 


SO 


16.2 



48 



4.9 


4- 


5-7 
6 5 


5- 
6. 


7-3 
8 i 


7- 
8 


.6.3 


16. 


24-5 


24 


32 6 


3« 


40-8 


40. 



48 4^ 



4-7 
5-5 
6.3 

71 

7-9 

'5-8 

23.7 

3»-6 
39-6 



19 

1.9 
3.2 

2-5 

2 8 
3-» 
6.3 

9-5 
12 6 

'5-8 



IB 



6 


1 


§ 


7 


2 


I 


8 


3 


4 


9 


3 


8 


10 


3 


X 


20 


6 


I 


30 


9 


3 


40 


13 


J 


50 


15 


4 



50 


49 


5-0 
5-8 
6.6 


4.9 

5-8 
6 6 


7-'; 
8.3 


7-4 
8.2 


.6.6 


16.5 


25.0 


24 7 


33 3 


33-0 


4'. 6 


41.2 



I', r. 



427 



TABLE VIIL— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

68° 09° 



10 

II 

12 



14 



15 
i6 

17 
i8 

19 



20 

21 

23 

24 



^3 

26 

27 
28 
29 



30 

31 

32 
33 

34 



35 

36 
37 
38 
39 



40 

41 
42 

43 

44 



45 
46 
47 
48 

49 



50 

51 

52 
53 
54 



55 
56 

57 
58 
59 



60 



Losj. Ters.' I> Loe. Exsec; D 



9.7961S 
•79634 

•79653 
.79671 
.79690 



9.79709 
.7972? 

•79746 
.79765 
•797S3 



9.79802 
.79821 

•79839 
.79858 
.79877 



9.79893 
.79914 
■79933 
•79951 
.79970 



9^79988 
. 80007 
. 80026 
.80044 
.8006^ 



9.80081 
.80106 
.80119 
.8013^ 
.80156 



9.80174 
.80193 
,80211 
.80230 
. 80248 



9.80267 
.80286 
. 80304 
•80323 
.80341 



9.80360 
•80378 
.80397 
.80415 
■ 80434 



9.80452 
. 80470 
. 80489 
.8050^ 
.80526 



9.80544 
.80563 
.80587 
.80600 
.80618 



9.80636 
.80655 
•80673 
. 80692 
.80710 



9.80728 



Log. Vers. 



18 
19 
18 
18 

19 
18 
19 
18 
18 

19 
18 
18 
19 
18 

18 
18 
19 
18 
18 

18 
19 
18 
18 
18 

18 
19 
18 
18 
18 
18 
18 
18 
18 
18 

19 
18 
18 
18 
18 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
J8 
18 
18 

18 

18 



10 



.222581 
.223081 
.223581 
.224081 
.22458 



10 



225081 

22558! 
22608 
22658 
227O8' 



10. 



22759 
22809 
22859 
22909 
22960 



10. 



23010 
23066 

231 15 
23161 
23211 



10 



23262 
23312 
23362 

23413 
23463 



10. 



23514 
23564 
23615 
23666 
23716 



10 



23767 
23817 
23868 
23919; 

239691 



10. 



24020 
24071 
24122 
24172 
24223 



10. 



24274 
24325 
24376 
24427 
24478 



10. 



24529 
24580 
24631 
24682 

24733 



10. 



24784 

24835 
24886 

24937 
24988 



10. 



25039 
25096 
25142 

25193 
25244 



10.25295 



50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
51 
50 
50 
51 
50 
51 
50 
51 
51 
50 
51 
51 
51 

51 
51 
51 
51 
51 

51 
51 
51 

51 
51 

51 
51 
51 
51 
51 
51 



Ters.! D 



80728 

80747 
80765 
80783 
80802 



80826 
80839 
80857 
80875 
80S94 



80912 
80936 
80949 
80967 
80985 



003 
022 
046 

058 
077 



095 
113 
131 
150 
168 



186 
204 
223 
241 
259 



27^ 
295 

314 
332 
350 



368 

386 
405 

423 
441 



459 
477 
495 
513 
532 



550 
568 
586 
604 
622 



646 
658 
6/6 
695 
713 



72)^ 
749 
767 
785 
803 



8182T 



Log. Exseo.; jf> 



10 



25295 
25347 

25398 
25449 
25501 



10. 



25552 
25604 

25655 
25707 
25758 



10 



25810 
25861 

25913 
25964 
26016 



10 



2606^ 
26 II 9 
2617 1 
26222 
26274 



10 



26326 
26378 
26429 
26481 
26533 



10 



26585 
26637 
26689 
26741 
26793 



7> 'Log. Kxsec i D ' Log. Vers.' 7) 

428 



10 



26845 
26897 
26949 
27001 

27053 



10 



,27105 
,2715^ 
, 27209 
27261 
27314 



10 



27366 

274I8 
27476 
27523 

2757? 



10 



2262^ 
27680 
27732 
27785 
2783? 



10. 



27890 
27942 
27995 
2804^ 
28100 



10.28152 
.28205 
.28258 
.28316 
.28363 



10.28416 



51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 

51 
52 
51 
51 

52 

51 
52 
51 
52 
52 

52 
51 

52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 
52 
52 
52 
52 
52 

52 
52 
52 
52 
52 
52 
52 
52 
52 
52 

5- 
53 
52 
52 
53 
52 



Log. Exsec. 1 /> 







10 

1 1 

12 

13 
14. 



15 
16 

17 
18 

19 



20 

21 

22 

23 

24 



p. P 



35 
36 
37 
38 
39 



40 

41 
42 

43 
44 



45 
46 
47 
48 
49 



50 

51 
52 

53 
54 



55 
56 
57 
58 
59 



60 



40 
50 



20 
30 
40 
50 



20 

30 
40 

50 



20 

30 
40 

50 



53 52 



5.3 
6.2 

7.0 

7.9 

8.8 
17-6 
26.5 

35-3 
44.1 



52 

5.2 

6.0 

6. a 
7.8 

8.6 
17^3 
26.0 

34-6 
43 3 



51 

51 

5-9 
6.8 

7-6 

8.5 

17.0 

25.5 
34-0 
42.5 



5.2 
'3.1 
7.0 
7.0 
8.7 

17-5 
26.2 
SS-o 
43'7 



51 

5-1 
6.0 

6-8 
.7-7 
8.6 

25-7 
34^3 
42.9 



50 

5-0 
5^0 
6.7 
7.6 
8.4 
16.8 
25.2 

33- 6 
42.1 



20 

30 
40 

50 



50 

5-0 



16.6 
25.0 

33.3 
41-6 



19 


I 


1.9 


1. 


2.2 


2. 


2-5 

2-§ 


2. 
2. 


3-1 
6.3 


3^ 
6. 


9-5 
12.6 


0. 
12. 


^5-1 


^5- 



18 



7 


2. 


8 


2. 


9 


2. 


10 


3- 


20 


6. 


30 


9^ 


40 


12. 


50 


15- 



P. p. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



o 



71 



i6 

18 
19 



20 

21 

22 

23 

24 



-5 
26 

27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



50 

51 

52 
53 
54 



:)5 
56 
hi 
58 
59 



(>0 



Lojf. Vers. ' I> 



9.8182I 
.81839 
.81857 
.81875 
.81893 



9 . 8 1 9 1 1 
.81929 
.81947 
.8196I 
.81983 



9.82001 
.S2019 
.82037 
.82055 
.82073 



9.82091 
.82109 

.82127 

.82145 
.82163 



9.82449 
.82467 
.82485 
.82503 
.82526 



9-82538 

•82556 
•82574 

•82592 
.82609 



9.82627 

•82645 
.82663 
.82681 
.82698 



9.82716 
•82734 
.82752 
.82769 
.82787 



9.82805 
.82S23 
.82840 
.82858 
.82876 



9.82181 
.82199 
.82217 
•82235 



9.8.^276 
.82288 
.82306 
■82324 
.82342 



9.82360 
.82378 
• 82396 
.82413 
.82431 



9. 82894 



liOs;. Vers. 



liOtf. K.xsec 



It 



10. 28416 
. 28469^ 
.28521 
.28574 
.28627 

28680 

28733' 
28786 
28839' 
28892 



10 



10 



28945 
28998: 
2905 I \ 
29104 
29157 



10. 



29210 
29263 

293161 
29370 
29423 



10. 



29476 
29529 
29583 
29636 
29689 



10. 



29743 
29796 
29850 
29903 
29957 



10. 



30010 
30064 
30117 
3o>7i, 

^0225i 



10 



30278| 
30332 
30386 

30440 
30493 



10 



30547 
30601 

30655 
30709 
30763 



10. 30817 
.30871' 
•30925 
• 30979 
•31033 



10 



31087 
31141, 
31195 
31249 
313031 



10 



31358; 
31412 

31466 
31521 

3'575i 



10. 31629 



53 
52 
53 
53 
52 
53 
53 
53 
53 
53 
53 
53 
53 
53 

53 
53 
53 
53 
53 
53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
54 

53 

53 
54 

r ■-> 

DJ 
54 
53 
54 
53 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 

54 
54 
54 
54 
54 
54 



Lot. Vers. 



/> 



/> |Loc. Kxsec. /> 



82894 
8291 I 
82929 
82947 
82964 



82982 
83000 
83017 
83035 
83053 



83076 
83088 
83106 
83123 
83141 



83159 
83176 
83194 
83211 
83229 



83247 
83264 
83282 
83299 
83317 



83335 
83352 
83370 

83405 



83422 
83440 
83458 
83475 
83493 



83510 
83528 

83545 
83563 
83586 



83598 
83615 

83633 
83656 

83668 



83685 

83703 
83720 

83737 
83755 



83772 
83790 
83807 
83825 
83842 



83859 
83877 
83894 
83912 

83929 



83946 



I.OL'. K\M-C.I 7> 



10 



10 



10 



lioe. Vers. 7> 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



31629 
31684 

31738 
3 ' 793 
31847 



31902 

3'956 
3201 1 
32066 
32126 



32175 
32230 
32284 
32339 
3239-1 



32449 
32504 

32558 
32613 
^2668 



32723 
32778 
32833 
32888 
32944 



32999 
33054 
33' 09 

33164 
33220 

33275 
33330 
33385 
33441 
33496 



33552 
3360^: 
33663 
33718 
33774 



33829 

33885 

33941 

339961 

34052! 



34108 

34164; 
342201 

34275J 
34331 1 
34387 
34443 
34499 
34555 
34611 

3466^ 

34723 
34780, 

34836I 
348q2 



34948! 



54 
54 
54 
54 

54 
54 
54 
55 
h\ 
54 
55 
54 
55 
hi 

55 
55 
54 
55 
55 
55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
56 
55 
55 
56 

55 
56 
56 
55 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 
56 



5 
6 

7 
8 

_9_ 
10 

1 1 

1 2 

'3 
14 



15 

16 

17 
18 

19 



r. r 



21 

22 

23 
24 



26 

27 
28 
29 

30 



34 



j3 
36 
37 
38 
39 



40 

41 
42 
43 
ii_ 

45 
46 

47 
48 

49 
oO 

51 
52 
53 
54 



^\^»M• 



/> 



55 
56 
57 
58 
59_ 

(;o 



20 

40 
50 



40 
50 



20 
30 
40 

50 



40 
50 



56 56 



5 6 
6.6 

7-5 
8.5 
9.4 
18.5 
28.2 
37-6 
47-» 



5.5 
6.5 
7-4 
8.3 
9.2 
T8.5 
27.7 

370 
46.2 



54 



53 

5-3 

6.2 

7-i 
8 o 
8.9 

26.7 

35-6 
44.6 



5.6 
6.5 
7^4 
8.4 
9 3 
i3.6 
28.0 

37-3 



5S 55 



5-5 

6.4 

7-3 

8.2 

9.1 

18.3 

27.5 

36$ 

45-8 



54 

4 



5 


4 


5- 


6.3 


6. 


7 


2 


7 


8 


2 


8. 


9 


r 


9 


18 


I 


18. 


27 


2 


?!• 


36 


3 


3^'- 


45 


4 


45- 



53 

5-3 
6.2 
7.6 

7-2 
8.| 

17-6 
26.5 

35-1 
44 I 



55 



6 


5-2 


7 


O.i 


8 


7.0 





7-9 


10 


8.7 


20 


'7-5 


30 


a6.2 


40 


35-0 


50 


43-7 



18 17 17 



2.4 

2-7 

3-0 
6.0 

9.0 



5-8 

8.7 

ii.fi 

14 .6 



'•7 



I', r 



42Q 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



10 

II 

12 

13 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 
29 



30 

31 

32 

33 

34 



35 
36 

37 
38 
39 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 

57 
58 
59 



60 



Loff. Vers. 



9-83946 
83964 
83981 

83999 
84016 



84033 
84051 
84063 
84085 
84103 



84126 

8413^ 
84155 
84172 
84189 



84207 
84224 
8424T 
84259 
84276 



84293 
84316 
84328 

84345 
84362 



84380 

84397 
84414 

84431 
84449 



D Los. Kxsec 



84466 

84483 
84506 

8451^ 
84535 



84552 
84569 

84586 
84603 
84626 



84638 
84655 
84672 
84689 
84706 



84724 

84741 

84758 

84775 
84792 



84809 
84826 
84844 
84861 
84878 



84895 
84912 
84929 
84946 
84963 



9.84986 



Log. Vers.! J> 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



Log 



34948 

35005 
35061 

35ii? 
35174 



35230 
35286 
35343 
35399 
35456 



35513 
35569 
35626 

35683 
35739 



35796 

35853 
35910 

35967 
36023 



36086 
36137 
36194 
3625T 

36308 



36366 

36423 
36480 

36537 
36594 



36652 
36709 
36766 
36824 
36881 



36938 
36996 

37054 
3711T 
37169 



D 



37226 
37284 
37342 
37399 
3745? 



37515 
37573 
37631 
37689 

37747 



37805 
37863 
37921 

37979 
38037 



38095 

38153 
38212 
38276 
38328 



38387 



56 
56 
56 
56 

56 
56 
56 
56 

57 

56 
56 
56 
57 
56 
57 
56 
57 
57 
56 
57 
57 
57 
57 
57 

SJ 
57 
57 
Si 
57 

57 
57 
Si 
Si 
Si 
57 
57 
58 
Si 
Si 

si 
si 

58 

57 
58 

58 
Si 
58 
58 
58 

58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 



Log. Vers. 



84986 

8499? 
85014 

85031 
85049 



85066 
85083 
85100 
85117 
85134 



85151 
85168 
85185 
85202 
85219 



85236 

85253 
85270 
85287 
85304 



85321 
85338 

85355 
85372 
85389 



85405 
85422 

85439 
85456 
85473 



85496 
85507 
85524 
85541 
85558 



85575 
85592 

85608 
85625 
85642 



85659 
85676 

85693 
85710 

85726 



85743 
85766 

85777 
85794 
85811 



8582? 

85844 
85861 
85878 
8^895 



8591 I 
85928 

85945 
85962 

85979 



85995 



£> Log 



l^lxsec! 7> i Lo:r. Vers.' 7> 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



liOi 



Exsec 



38387 
38445 
38504 
38562 
38621 



38679 
38738 
38796 
38855 

389'4 



38973 

39031 
39096 

39149 
39208 



39267 
39326 
39385 
39444 
39503 



39562 
3962T 
39681 
39740 
39799 



39859 
39918 

3997? 
40037 

40096 



40136 
40216 
40275 
40335 
40395 



40454 
40514 

40574 
40634 
40694 



40754 
40814 

40874 
40934 
40994 



41054 
41114 

41 '74 
41235 

41295 



j> 



41355 
41416 

41476 
41537 
41597 



41658 
41719 

41779 
41840 
4 1 90 1 



41962 



58 
58 
58 
58 

58 
58 
58 
59 
58 

59 
58 
59 
59 
58 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
60 

59 
59 
60 

59 
60 

59 
60 

60 

60 
60 
60 
60 
60 

66 
60 
60 
66 
66 

60 
66 
60 
66 
66 

66 
61 
66 
66 
61 
61 



Kxsec. /> 



15 
16 

17 
18 

19 



20 

21 
22 

23 
24 



35 
36 
37 
38 
39 



40 

41 
42 

43 
44 



45 
46 
47 
48 
49 



50 

51 
52 

53 
54 



55 
56 
57 
58 
59 



00 



P. P. 



40 
50 



40 

53 



40 
50 



6 

7 

8 

9 
10 
20 
30 
40 
50 



20 
30 
40 

50 



6 


I 


6.1 1 


7 

8 


I 
I 


9 
10 


1 
i 


20 


3 


30 


5 


40 
50 


6 
§ 



60 

6.0 
7.0 
8.0 
9.0 

10.0 
20.0 

30.0 

40.0 

50.0 



59 



5-9 
6.9 


5- 
6. 


7-8 
8-8 


7- 
8. 


9-8 


9- 


J9-6 


39- 


29-5 


29. 


39-3 
49.1 


39 
48. 



58 

5-8 
6.7 
7-7 
8.7 
9-6 
19-3 
29.0 

38.6 
48.3 



5-7 
6.6 
7.6 

8.5 
9-5 

IQ.O 
28.5 
38.0 

47-5 



65 

6.0 
7.0 
8.0 
9.1 

10. T 
20.1 
30.2 

40- 3 
50.4 



59 

5-9 
6.9 

7 9 
8.9 
9,9 

i9-§ 
29.7 

39-6 
49.6 



58 



57 

5-7 
6.7 

7-6 
8.6 
9.6 
19. 1 
28.7 
38.3 
47-9 



57 56 





17 


17 


6 


1-7 


1-7 


7 


2 





2.0 


8 


2 


3 


2.2 


9 


2 


6 


2.5 


10 


2 


Q 


2-8 










20 


S 


X 


5 6 


30 


8 


7 


8.S 










iO 


II 


6 


1^-3 


50 


14 


6 


14.1 



5-6 
6.6 

7-5 
8.5 
9-4 
38.8 
28.2 
37-6 
471 



16 

1-6 

1.9 



137 



P. P 



430 



TABLE VIII. — LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



7 4 



7 a 



10 

II 

12 



14 



15 
i6 

17 
i8 

19 



20 

21 

2 2 

24 



25 

26 

27 
28 

29 



30 

31 
32 
33 
34 



35 
36 

37 
38 

39 



40 

41 

42 
43 
44^ 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



55 
56 

57 
58 
59 



00 



Lost. Vers. 



9.85995 
.86012 
.86029 
. 86046 
. 86062 



9.86079 
. 86096 
.86113 
.86129 
.86146 



7> Log 



9.86163 
.86179 
.86196 
.86213 
.86230 



9.86246 
.86263 
.86280 
.86296 
.86313 



9.86330 
• 86346 
.86363 
.86380 
■86396 



9.86413 
. 86430 

• 86446 
. 86463 

• 86479 



86496 

86513 
86529 

86546 

86s62 



•86579 
.86596 
.86612 
.86629 
.86645 



9.86662 

.86678 
.86695 
.86712 
•86728 



9.86745 
.86761 
.86778 
.86794 
.86811 



9.86827 
. 86844 
. 86866 
.86877 
.86893 



9.86910 
.86926 
. 86943 
.86959 
. 86976 



9.86992 



Lost. Vers. 



17 

16 
17 
'6 
17 
16 
17 
16 
17 

1(5 
16 
17 
16 
17 

16 
16 
17 
16 
16 
17 
'6 
16 
'7 
16 

16 
17 
16 
16 
16 
17 
16 
16 
16 
16 
17 
16 
16 
16 
16 
16 
16 
17 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
'6 
16 
16 
16 
16 
16 



10 



/> Lo:r 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



Exsecj 1) 



41962 
42022 
42083 
42144 
42205 



42266 
4232? 

42388 
42450 
42511 



42572 
42633 
42695 
42756 
42817 



42879 

42940 
43002 
43063 
43' 25 



43 '87 

43249 
43310 
43372 
43434 



43496 
43558 
43620 
43682 
43744 



43806 
43868 
43931 
43993 

44055 



44118 
44180 
44242 

44305 
44368 



44430 
44493 
44556 

446 1 8 
4468 T 



44744 
44807 
44870 

44933 
44996 



45059 
45122 

45185 

45248 
45312 



45375 
45439 
45502 

45565 
45629 



45693 



Kxspc. 



60 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 

6! 
61 
61 

61 
61 
61 
61 
62 
61 
62 
61 
62 
61 

62 
62 
62 
62 
62 
62 
62 
62 
62 
62 

62 
62 
62 
62 

63 
62 
62 

63 
62 

63 

62 

63 
63 
63 
63 
63 
63 
63 
63 
63 

63 
63 
63 
63 
63 
64 



Lojf. Vers. /> 



9.86992 
. 87009 
.87025 
.87042 
.870 58 

9.87074 
.87091 
.87107 
.87124 
.87146 



9.87157 

.87173 
.87189 
.87206 
.87222 



9.87239 

.87255 
.8727! 

.87288 
• 87304 



9.87326 
.87337 
•87353 
.87370 
.87386 



9.87402 
.87419 

.87435 
.87451 

.87468 



9.87484 
.87506 

.875I6 
•87533 
.87549 



9.87565 
.87582 

.87598 
.87614 
.87631 



9.87647 
.87653 

.87679 
.87696 
.87712 



9.87728 
.87744 
.87761 

.^7777 
.87793 



9.87809 
.87825 
.87842 
.87858 
.87874 



9.87896 

. 87906 
.87923 

•87939 
.87955 



9.87971 



K\s»'o. 



10 



10 



•45^93 
•45756 
.45820 
.45884 
.45947 
.4601 1 
.46075 

•46139 
.46203 
.4626^ 



10 



•46331 

• 46395; 
. 46460 
.46524' 
.46588 



10 



,46652 
,46717 
,46781 
46846 
46916 



10 



•46975 
• 47040 
.47104 
,47169 
.47234 



10 



.47299 

•47364 
.47429 

■47494 
■47559 



10 



.47624 
,47689 

.47754 
,47820 
47885 



10 



.47950 

.48016, 

.48081 

.48147 

•48213 



10 



•48278 
.48344 
.48410 
.48476 
,48542 



10 



.48607 
.48674 
■ 48740; 
.48806, 
.48872 



10 



.48938 
. 49004 
,49071 

.4913?, 
,49204! 



10 



,49270 
.49337 
49403 
49476 

49537 



10.49604 



7> i liO tf. V p r s . I J) IliO g . Kx»<er.' 



I*, r 





67 


66 


6 


6.7 


6.^ 


7 


7.8 


7.7 


8 


8.9 


8.8 


q 


10. 


10. 


10 


II. I 


II. I 


20 


■22.3 


22.1 


30 


33.5 


33.2 


40 


44.6 


44-3 


50 


55.8 


554 





65 


65 


6 


6. .5 


6.5 


7 


7-6 


7.6 


8 


87 


«f? 


9 


9.8 


9.7 


10 


10. 9 


10. 8 


20 


21. g 


21.6 


30 


32-7 


32.5 


40 


43-6 


43-3 


50 


54^b 


54 I 



66 

6.6 

7.7 
8.8 

9.9 

I I o 



33.0 
44.0 
55.0 



64 

6.4 





64 


63 


63 


6 


6.4 


6.3 


6.3 


7 


74 


7-4 


73 


8 


8.5 


8.4 


8.4 


9 


9.6 


9 5 


9.4 


10 


10.6 


10.6 


JO-5 


20 


21.3 


21 . 1 


21 .0 


30 


32 


3'.7 


31-5 


40 


42-6 


42.3 


42.0 


50 


53 3 


52-9 


52.5 



62 62 61 



6 


6.2 


6.2 


6. 


7 
8 


7.3 
8.3 


7.2 
8.2 


7 
8 


9 
10 


9.4 
10.4 


9-3 
10.3 


9- 
10. 


20 


20. § 
31.2 
4J-6 

52.1 


20.6 


20. 


30 
40 

50 


31.0 
41 3 
5i^6 


30. 

4t 

5'. 



20 

30 
40 

50 



61 

6.1 

7-1 
8.1 

Q.I 

10. i 
20.3 
30-5 
40 § 
50-8 





17 


16 


6 


1-7 


^•6 


7 


2.0 


19 


8 


2.2 


2.2 


9 


2-5 


2.5 


10 


2.3 


2-7 


20 


S-6 


5-5 


30 


8.5 


8.5 


40 


II. 3 


1 1,0 


50 


14.1 


13.^ 



66 

6.5 
7.5 

8.5 

9.1 

10. 1 

20.1 

30.2 

40 3 
50.4 



16 

1.6 



2.^, 

8.0 
10.^ 
»3-3 



P. P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 







Lop. Vers. 



10 

II 

12 

14 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 
42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 

57 
=;8 

59 



(JO 



9.87971 
.87987 
. 88003 
. 88020 
.88036 



9.88052 
. 88068 
. 88084 
.88100 
.88116 



9.88133 
.88149 

.88165 
.88181 
.88197 



9.88213 
.88229 

.88245 
.88261 
.8827^ 



9.88294 
.88310 
.88326 
.88342 
.88358 



9.88374 
.88390 
. 88406 
.88422 
.88438 



9.88454 
.88470 
.88486 
.88502 
.88518 



9.88534 
.88556 
.88566 
.88582 
.88598 



9.88614 
.88636 

.88646 
.88662 
.88678 



9.88694 
.88710 
.88726 
.88742 
.88758 



9,88774 
.88790 
.88805 
.88821 
.8883^ 



9.88853 

.88869 

.88885 

88901 

.88917 



9.88933 
Lojj. Vers. 



J> 



6 
6 
6 
6 
6 

6 
6 

6 
6 
6 
6 

$ 
6 
6 
6 

"TT 



liOP 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



Kxsec. D 



49604 
49670 

4973? 
49804 
49871 



49939; 
50006' 
50073 
50146 
50208; 



50275 

50342 
50410 

5047? 
50545 



50613 
5068 1 j 

50748; 
508 1 6 
50^ 



50952 
51026 
51088 

51157 
51225 



51293 
51361 

51430 
51498 
51567 



51636 
51704 

51773 
51842 
51911 



51980 
52049 
52118 
52187 
52256 



52325 
52394 
52464 

52533 
52603 



52672 
52742 
52812 
52881 
52951 



53021 

53091 
5316T 

53231 
53301 



53372 
53442 
53512 

53583 
53653 



53724 ' 

Kxsec. 



66 
67 
67 
67 

6? 
67 
67 
67 
6f 

67 
6f 
6f 
67 
68 

6f 
68 

6? 
68 
68 

68 
68 
68 

68 
68 

68 
68 

68 

68 

68 

69 

68 

68 

69 

69 

69 

69 
6q 

69 
69 

69 
69 

69 
69 
69 

69 
70 
69 

69 
70 

70 
70 
70 
70 
70 
76 
70 
76 
70 
76 
70 

"TT 



Los. Vers. 



88933 
88949 
88964 
88986 
88996 



89012 
89028 
89044 
89060 
89075 



89091 
8910^ 
89123 

89139 
89155 



89176 
89186 
89202 
89218 
89234 



89249 
89265 
89281 
89297 
893 • 2 



89328 
89344 
89360 

89376 
89391 



89407 
89423 
89438 
89454 
89470 



89486 
89501 
8951^ 
89533 
89548 



89564 
89580 
89596 
896 11 
89627 



89643 
89658 
89674 
89690 
89705 



89721 

89737 
89752 
89768 
89783 



89799 
89815 
89836 
89846 
89862 



9.8987? 



liOsr. Vers. 



U Lo;r. Exsec. JJ 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



T) ihuii 



53724 

53794 
53865 
53936 
54007 



54078 

54149 
54220 
54291 
54362 



54433 
54505 
54576 
5464? 
54719 



5479' 
54862 

54934 
55006 
55078 



55150 
55222 

55294 
55366 
55438 



5551^ 
55583 

55655 
55728 
55801 



55873 
55946 
56019 
56092 
56165 



56238 
563 1 T 

56384 
5645? 
5653' 



56004 
56678 

56751 
56825 

56899 



56973 
57047 
57126 

57195 
57269 



57343 
5741? 

57491 
57566 
57646 



57715 
57790 
57864 

57939 
58014 

58089 

Kxsec. T> 



70 

71 

76 

71 

71 
71 
71 
71 
71 

7T 
71 
71 
71 
72 

71 
71 

72 
71 

72 
72 
72 
72 
72 
72 

72 
72 
72 

73 

72 

72 

73 
72 

73 
73 

73 
73 
73 
73 
73 
73 
73 
73 
74 
73 

74 
74 
73 
74 
74 
74 
74 
74 
74 
74 

75 
74 
74 
75 
75 
75 



I 



25 
26 

27 

28 



ao 

31 
32 
33 

34 



35 
36 

37 
38 
39 



40 

41 
42 

43 
44 



45 
46 
47 
48 
49 



P. P. 





75 


74 


72 


t 


7.-S 


7-4 


7. 


7 


8.7 


8 


6 


8. 


8 


10. 


9 


8 


9 


9 


II. 2 


II 


I 


10. 


10 


12.5 


12 


3 


12. 


20 


25.0 


24 


6 


24. 


30 


37.5 


37 





3b. 


40 


50.0 


49 


3 


48. 


50 


62.5 


61 


6 


60. 



20 

30 
40 
50 



66 


6.6 1 


7 


7 


8 


8 


9 


9 


1 1 





22 





33 





44 





55 






16 



13 



16 

T.6 



2.4 
•^•6 
5-3 
8.0 
10. 6 
J3-3 





72 


71 


6 


7.2 


7.1 


7 


8.4 


8.3 


8 


9.6 


9.4 


9 


10.8 


10. 6 


10 


12 .0 


".8 


20 


24.0 


23-6 


30 


36.0 


35.5 


40 


48 


47.3 


50 


60.0 


59-1 



70 

7.0 

8.2 

9.4 

10.6 

11.7 
23.3 

35-2 
47.0 
58.7 



69 68 67 



6 


6.Q 


6.8 


7 


8,5 


7-9 


8 


9.2 


9.0 


9 


10.3 


10 2 


10 


"5 


".3 


20 


2:!.0 


22-6 


30 


34. s 


34 


40 


46.0 


45-3 


50 


57.5 


50-6 



6.- 
7.8 
8.9 
100 
II. i 
2P.3 
33-5 
44-6 
55.8 



0.0 
0.0 
0.0 
o. 1 

O. I 

o. i 
0.2 

0.3 
0.4 



15 



P. P. 



432 



TABLE Vlll. — LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



78' 



ly c 



1) 



Lo?. Vers. | 7> 






9.8987^ 


15 
I? 


10.58089 


I 


.89893 


.58164 


2 


• 89908 


•58239 


3 


.89924 


•58315 


4 


• 89939 


16 

, 3 


• 58390 


S 


9.89955 


10.58465 


6 


.89971 


, 2 


.58541 


7 


.89986 


I = 


•586I6 


8 


.90002 


. 58692 


9 


.90017 



t6 


.58768 


10 


9.90033 


10.58844; 


1 1 


• 90048 


.58920 


12 


. 90064 


.58995 


13 


. 90080 


I 2 


.59072 


U 


.90095 


I? 


.59148 
10.59224 


15 


9.901 1 1 


i6 


.90125 


I = 


.59300 


17 


.90142 


' 3 
I = 


•59377; 


i8 


•90157 


T » 


•59453 


19 


.90173 


•59530 


20 


9.90188 


10.59606 


21 


. 90204 


T £ 


•59683 


22 


.90219 


• 59760; 


23 


•90235 


^ 5 

1 = 


•59837, 


24 


.90250 


T » 


• 59914; 
10.59991' 


25 


9.90266 


26 


.90281 1 


' 5 

T P 


.60068 


27 


.902971 


I5 
, 3 


.60145 


28 


.90312 j 


'5 


.60223 


29 


.90328 ' 


.60306 


30 


9 •90343 


10.60378 


31 


•903591 


' 5 
T r 


.60455, 


32 


•90374 




•60533 


33 


.90389 




. 606 1 1 1 


34 


•90405 


Id 

, p 


.60688 


35 


9 . 90426 


10.60765 


36 


.90436 


I3 

T = 


. 60844 


37 


.90451 


I 3 

. P 


.60923 


3^ 


.90467 


I 5 
, p 


. 6 I 00 I 


39 


.90482 


I3 

15 


.61079 


40 


9.90497 


10.61158 


41 


•90513 


r P 


.61236 


42 


•90528 


r = 


.61315 


43 


•90544 


1 = 


•61393 


44 


•90559 


' 3 
15 


.61472 


45 


9.90574 


10.61551 


46 


.90590 


.61630 


47 


.9060 5 


, P 


.61709 


48 


. 9062 1 


'5 

1 p 


.61788 


49 
oO 


•90636 




15 
, p 


.6186^ 


9.90651 


10.61947 


51 


. 90667 


'3 


.62026 


52 


.90682 


I 3 


.62105 


53 


. 90697 




T = 


.62185 


54 


. 907 1 3 


T P 


.62265 


55 


9.90-28 


10.62345 


56 


• 90744 


15 
15 


.62424 


57 


•90759 


.62504 


5^ 


.90774 


.62585 


59 


. 90790 


.62665 


GO 


9.90805 


10.62745 


' 


liOtf. Vers. 


1 /> 


liOe. Kxsec. 



Loff. Kxsec. J> 



75 
75 
75 
71 

75 

75 

75 
76 

75 

76 
76 

75 
75 

76 
76 

76 
76 
76 
76 

76 
77 
76 
77 
77 

77 
77 
77 
71 
77 

77 
77 
77 
78 
77 
78 
78 
78 
78 
78 

78 
78 
78 
78 
79 

78 
79 
79 
79 
79 

79 

79 

79 
80 

79 
80 

79 
80 
86 
80 
80 

/> 



Loir. Vers. 



90805 
90826 

90835 
90851 

90865 



7> 



90881 
90897 
90912 
90927 
90943 



90958 
90973 
90988 
91004 
9IOI9 



91034 
91049 

9 1 06 5 
91080 
91095 



91 1 10 
91 126 
91141 

91156 
91171 



91 187 
91202 
91217 
91232 
91247 



91263 
91278 
91293 
91308 
91323 



91338 
91354 
91369 

91384 
9' 399 



91414 
91429 

9 '445 
91460 

91475 



91490 
91505 
91520 

91535 
91556 



91565 
91581 
91596 
9161 1 
91626 



91641 
91656 
91671 
91685 
91701 

9 • 7 ' r ) 



liOjr. Kxser. 



10 



62745 
.62825 
,62906 
,62985 
,6306^ 



10 



,63148 
,63229 
.63310 

.63391 
,63472 



10 



63553 
63634 
63716 

6379? 
63879 



10 



63961 
64043 
,64125 

,64207 
,64289 



1) 



10 



.64371 
64453 
•64536 
,64618; 
,64701! 



10 



10 



64784 
64867 

64950 

65033 
,651 16 

^5199 
65283 

65366 
,65450 

65534 



10 



6561^ 

,657oT 

.65785 
,65870 

•65954 



10 



66038 J 
,66123 
,6620^1 
.66292 
,66377 



10 



10 



10 



,66462 
.66547 
,66632 
.66717 
.66803 

"66888 
,66974 
.67059 
,67145 
.6723T 

673"? 

.67403 

67490 

.67576 
• 67663 



10,^7749 



I IjOtr. Vers. /> l,<iir- Kxser 

433 



86 
86 
81 
85 
81 
81 
81 
81 

81 
81 
81 
81 
81 

82 
82 
82 
82 
82 
82 
82 
82 
82 

83 
82 
83 
83 
83 
^1> 

83 
83 
83 

83 
84 

83 
84 

84 
84 
84 

84 
84 
84 
84 
85 

85 
85 
85 
85 
85 

85 

85 

85 
86 

86 

86 
86 

86 
86 
86 
86 



/> 



5 
6 

7 
8 

_9_ 
10 

1 1 
12 

13 
14 



15 
16 

17 
18 

19 



20 

21 

22 

23 
24 



25 
26 

27 
28 
^9 

30 

31 
32 
33 
34 



35 
36 
37 
38 

39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

50 

5' 
52 
53 
54 



55 
56 
57 
58 
59 



r»o 



1'. I' 



86 85 84 



6 


8.6 


8.5 


7 


10. 


9 


8 


11.4 


11.3 


9 


12. g 


12.7 


10 


14.3 


14. 1 


20 


28.6 


28.3 


^0 


43.0 


42.1; 


40 


57-3 


56.6 


50 


7»^6 


70.8 



8. 
9.8 

II .2 

12.6 

14.0 

28.0 

42.0 

56.0 
70.0 



83 82 81 



6 


8.3 


8.2 


8. 


7 


9-7 


9-5 


9- 


8 


11 .0 


10. 2 


10. 


9 


12.4 


12.3 


12. 


10 
20 


i3-§ 
27-6 


X3-6 
27-3 


'3- 
27. 


30 


41. s 


41.0 


40 


40 
50 


55-3 
69.1 


54 § 

68.3 


54 • 
67 





80 


79 


6 


80 


7-9 


7 


9-3 


9.2 


8 


10.6 


10.5 


9 


12.0 


"•§ 


10 


133 


13-1 


20 


26.6 


26.3 


30 


40.0 


39-5 


40 


53-3 


52-6 


50 


66.6 


65-8 



78 

78 

9.1 

10.4 

11.7 
13.0 

26.0 
39-0 
52.0 
65.0 



77 76 75 



6 


7^7 


7.6 


7 


9.0 


8.8 


8 


10.2 


10. 1 


9 


"•5 


II. 4 


10 


«« § 


12. $ 


20 


25 6 


253 


30 


38.5 


38.0 


40 


5'-3 


50. § 


50 


64.1 


63.3 



7-5 
8.7 
10. o 
1 1 .2 
12.5 
25.0 

37-5 
50.0 
62.5 



20 
30 
40 

50 



0.3 
0.4 





16 


15 


6 


1.6 


i'5 


7 
8 


3.1 


1.8 
2.6 


9 


2-4 


23 


10 


2.5 


2.6 


20 


5-3 


5^» 


30 


8.0 


7-7 


40 


10$ 


10.3 


50 


133 


12.9 



15 

'•5 
'•7 
2.0 
2.3 

2-5 

7-5 
to.o 

12.5 



I', r 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



80^ 



81 



10 

1 1 

12 

14 



15 

i6 

17 
i8 

19 



20 

21 

22 

23 
24 



25 
26 

27 



29 

30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 

46 

47 
48 

49 



50 

51 
52 
53 
54 



55 
56 

57 
58 
59 



60 



Log. Vers 



7I6 
731 
746 
761 

776 



791 
807 
822 
837 
852 



867 
882 
897 
912 
927 



942 

957 
972 

987 



92002 



92016 
9203! 

92046 
92061 

92076 



92091 
92106 
92121 

92136 

92151 



92166 
921 81 
92196 
9221 r 
92226 



92240 
92255 
92276 
92285 
92306 



92315 
92330 

92345 
92360 

92374 



92389 
92404 
92419 

92434 
92449 



92463 
92478 
92493 
92508 

92523 



92538 
92552 
9256^ 
92582 
92597 



9.92612 



Log. Vers. 



X) Loff. Exsec. 



10 



67749 
67836 

67923 
.68010 
. 68097 



10 



,68184 
.6827 

68359 
.68447 

.68534 



10 



,68622 
68716 
,68798 

,68886 
68975 



10 



69063 
,69152 
69246 
69329 
694 1 8 



10 



69507 
69596 
69686 

69775 
69865 



10 



69955 
, 70044 

■70134 
,70224 

.70315 



10 



70405 
70495 
70586 
70677 
70768 



10 



70859 
70950 
7104T 

71133 
71224 



10 



,71316 
,71408 
7 1 500 
71592 
,71684 



10 



71776 
71869 

7 1 96 1 

72054 

72147 



10 



,72240 

72333 
72427 
,72526 
72614 



10 



,7270^ 
7280T 
72895 
72990 
73084 



10.73178 



/> Log. Kxsec. 



I) 



86 

87 
87 
^7 
8^ 
8? 
87 
8? 
87 
88 



88 



89 
89 

89 
89 
89 

89 
89 
90 
89 
90 
90 
96 

90 

90 

91 
96 

91 

91 
91 
91 
91 
91 

91 
92 
92 

92 
92 

92 
92 
92 

93 

92 

93 
93 
93 
93 
93 

93 
94 
94 
94 
94 
94 



Log. VerS' 



I) 



92612 
92626 
92641 
92656 
92671 



92686 
92706 
92715 
92730 
92745 



92759 

92774 
92789 
92804 
92818 



92833 
92848 
92862 
9287^ 
92892 



92907 
9292T 
92936 
92951 
92965 



92986 
92995 
93009 
93024 

9.3039 



93053 
93068 
93083 

9309^ 
93II2 



93127 

9314I 
93156 
9317I 
93185 



93200 

93214 
93229 

93244 
93258 



J> Log. Exsec. 



93273 
93287 

93302 

93317 

93331 



93346 
93366 

93375 
93389 
93404 



93419 
93433 
93448 
93462 

93477 



9.93491 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



Loe. V«*rs.l 7> |Loi 



73178 
73273 
73368 

73463 
73558 



73653 

73748 
73844 
73940 
74035 



74131 
7422^ 

74324 
74426 

74517 



74613 
74716 

7480^ 

74905 
75002 



ij 



75099 
7S^9l 
75295 
75393 
7549' 



75589 
75688 

75786 
75885 

75984 



76083 
76182 
76282 
76382 
76481 



76581 
76681 
76782 
76882 
76983 



77083 
77184 
77286 
77387 
77488 



77590 
77692 

77794 
77896 
77998 



78101 
78203 
78306 
78409 
78513 
78616 
78720 
78823 
78927 
79031 



79136 



Exsec. 



95 
94 
95 
95 
95 
95 
95 
96 
95 
96 
96 
96 
96 
96 
96 
97 
97 
97 
97 

97 
98 

9l 
98 

98 

98 
98 
98 
99 
99 

99 

99 

99 

100 

99 
00 
00 
06 
06 
06 

06 
01 
01 
01 
01 

01 
02 
02 
02 
02 
02 
02 
03 
03 
03 

03 
04 

03 
04 
04 
04 



/> 



10 

I 

2 

3 

4 



20 

21 

22 

23 
24 



25 
26 

27 

28 

29 



30 



34 



35 
36 
37 
38 
39 



40 

41 

42 
43 
44 



45 
46 
47 
48 
49 



50 

51 
52 

53 

54 



55 
56 
57 
58 
59 



«0 



p. r 



20 

40 
50 



6 

7 
8 

9 
10 
20 

30 
40 

50 



40 
50 



20 

30 
40 

50 



20 

30 
40 

50 



90 



9 





TO 


5 


12 





13 


5 


15 





30 





45 





60 





75 






0-5 
0.6 
o.g 
0.7 
o§ 
1-6 
2.5 
3-3 
41 



IS 



40 
50 






7 


0. 





8 








Q 


0. 


I 








I 


I 


1. 


2 


3 


2. 


3 
4 


5 
6 


3- 
4- 


5 


8 


5- 



80 

8.0 

9-3 



13-3 
26.6 
40.0 

53 3 
66.6 



0.8 
0.9 
1 .0 
1.2 
1-3 
2.6 
4.0 
5-3 
6.6 



IS 



I 


S 


1. 


I 


8 


I. 


2 





a. 


2 


3 


2. 


2 


6 


2. 


5 

7 

10 


I 
7 
.3 


5- 

7- 

10. 


12 


9 


12. 



14 

1.4 

1-7 
1.9 
2.2 
2.4 

4-8 
7.2 

9-6 
12. 1 



434 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

83° 8:r 



10 

II 

12 



15 
i6 

17 
i8 

19 



20 

21 
22 

23 

24 



25 
26 

27 

28 

29 



30 

31 

32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 

47 
48 

49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



L(>S. Vers. I 7> 



9-93491 
93506 



'J3- 



93535 
93549 



93564 
93578 
93593 
93607 
93622 



93636 
93651 
93665 
93680 
93694 



93709 
93723 
93738 
93752 

93767 



93781 

93796 
93816 
93824 
93839 



93^53 
93868 
93882 
93897 
9391 1 



93925 
93940 
93954 
93969 
93983 



93997 
94012 

94025 
94041 
940 s 5 



94069 
94084 
94098 
941 12 
94127 



94141 

94155 
94170 

94184 
94198 



94213 
94227 

94241 
94256 
94270 



94284 
94299 

94313 
9432? 
94341 
9 94356 

liOe. Vers. I 7> 



14 

u 
14 
14 

14 
14 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

u 
14 
u 
14 
14 

14 
14 

14 
14 
u 
14 
14 
14 
14 

14 
14 
14 

14 
14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 
14 



Log. Ex sec. I D \ 



IO.79F36J 
.792401 

•79345 
•79450 
•79555 



10.79666 
.79766 
.79871 
•79977 
. 80083 



10.80189 
. 80296 
. 80402 
.80509 
.806 16 



10.80723 
.50831 
•80938 
.81046 
.81154 



10,81202 
.813711 

.81479 
.8i:;88 
.81697 



io.8i8o6 
.81916 
.82025 
.82135 
.82245 



10.82356' 
.82466 
.82577 
.82688 
.82799 



10.82916 
.83022 

•83133' 
.83245 
• 833vS' 



10.83470 

.83583 
.83695 
.83809 
.83922 



10.84035 
.84149 
•84263 

. 84492 



10.84607 
.84721 
.84837 
.84952 
.85068 



10.85183 

.85299 
.85416 

•85532 

. 85649 

10.85766 



104 
105 
104 
105 

105 
105 
105 
106 
106 

106 
106 
106 
107 
107 

107 
I of 
\oJ 
108 
108 

108 
log 
log 
109 
109 

109 
109 
109 

1 10 
no 

no 
1 16 
116 

1 1 1 
III 
III 
1 11 
1 11 

I 12 
I 12 
112 
I 12 
I 12 
113 
113 

113 
114 
114 
114 
114 

U5 

114 

115 
116 

116 

116 
116 
117 
117 



Log. Vers. I /> L(»;r. Kxscc. 



Kxspr.l 7> 



9 -943 56 
94370 
94384 
94398 
94413 



94427 
94441 
94456 
94470 
94484 



94498 
94512 

94527 
94541 
94555 



945<^9 
94584 
94598 
94612 

94626 



94646 , 

94655 j 
94669 

94683 

94697 I 



947 1 1 
94726 

94740 
94754 
94768 



94782 

94796 
94816 
94825 
94839 



94853 
94867 
9488T 
94895 
94909 



94923 
94938 
94952 
94966 
94980 



94994 
95008 
95022 

95036 , 
95050 



95064 
95078 
95093 
95107 
95121 



95135 
95M9 
95163 
95177 
95191 
995205 

liOR. Vers. 7> 



10.85766 
.85884 
. 86001 
.861 19 
.86237 



/> 



10 



86355 
,86474 
,86592 
,86711 
,86831 



10 



.86956 
.87076 
.87196 
,87316 
.87431 



10 



87552 
87673 
87794 

.8;;9i6 
,88038 



10. 



88160 
88282 
88405 
88528 
8865T 



10 



88775 
,88898 
,89022 
.89147 
,89271 



10 



89396 
,89521 
,89647 

.89773 
.89899 



117 

iif 
117 
118 

118 
118 
118 
119 
119 

119 
120 

120 
120 
126 

121 
121 

121 

! 121 
122 

122 
122 
122 
123 
123 
124 
123 
124 
124 
124 
125 
125 
125 
126 
126 



10 



.90025 
, 90 1 5 2 
,90279 
, 90406 
90533 



10 



,90661 

.90789 
,90917. 

,91046 
91175 



10 



91304 
.91434 
.91564; 
.91694, 

91825, 



10 



91956 
,92087 

,92218 
.92350 
,92482; 



10 



92614 

92747 
92886 

93014' 

9314 71 
93281I 



126 
126 

127 
127 
127 

128 

127 

128 

129 

129 

129 

1 130 

129 

136 
130 

131 
131 

3 
3 



13" 
131 
132 

132 
133 
133 
133 
133 
134 



4 

5~ 
6 

7 
8 

9 
10 

1 1 
12 

13 
14 

15 
16 

17 
18 

19 



21 

23 
24 



10 



l.fijr. Kxsec' /> 



25 
26 

27 
28 

30 

31 
32 
33 
34 



I', r. 



35 
36 

37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 

:>o 

51 
52 
53 
54 



55 
56 
57 
58 
59 



<io 



6 


130 

13.0 


7 
8 


»5-J 
»7^3 


9 


19.5 


10 


2» f, 


20 

40 

50 


43 3 
65.0 

108.3 



120 

12. (J 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
£0 o 

lUO.O 



no 100 

10.0 
II. 6 
>3-3 
15.0 
16.^ 

33.3 
50.0 

66.6 

83.3 



6 


II .0 1 


7 


'2.^1 


8 


M.6 





16. s 


TO 


,8.3 


20 


36.6 


30 


55.0 


40 


73 3 


50 


91-6 



6 


3 

0.3 


7 

8 


0.3 
0.4 


q 


0.4 


10 


0.5 


20 


1 .0 


30 


1.5 


40 


2.0 


50 


2-5 





I 


6 


0.. 1 


7 





I 


8 





I 


9 





I 


10 





I 


20 





3 


30 





5 


40 





^ 


50 





8 





14 


6 


1.4 


7 


».7 


8 


» 9 


9 


2.2 


10 


a. 4 


20 


4.? 


30 


7.2 


40 


9-6 


50 


12. 1 



0.1 

O.I 

o.i 
o.a 
0.3 
0.4 



14 
1.4 
15 
» 8 
a. I 

2-3 

4 6 
7.0 

9.3 
11.6 



r. r. 



435 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
84° 85° 



10 

II 

12 



15 
i6 

17 
i8 

19 



20 

21 

22 

23 

24 



25 
26 

27 

28 

29 



30 

31 
32 
33 
34 



35 
36 

37 
38 
39 



40 

41 

42 

43 
44 



45 
46 

47 
48 

49 



50 

51 

52 
53 
54 



55 
56 
57 
58 
59 



60 



Lost. Vers. 



9.95205 
95219 

95233 
9524f 
9526T 



95275 
95289 

95303 
9531^ 
95331 



95345 
95359 
95373 
95387 
95401 



95415 
95429 
95443 
95457 
95471 



2> 



Log. Exsec. 



95485 

95499 
95513 
95527 
95540 



95554 
95568 
95582 

95596 
95610 



95624 
95638 
95652 
95666 
95680 



95693 
95707 
95721 

95735 
95749 



95763 

95777 

95791 
95804 

95818 



95832 

95846 
95860 

95874 
95888 



95901 

95915 
95929 

95943 
95957 



95970 
9598^ 

95998 
96012 
96026 



9.96039 



Log. Vers. 



n 



10 



93281 
93416 

•93551 
93686 

93821 



10 



93957 
.94093 
,94229 

94366 
94503 



10 



,94641 
94778 
.94917 
.95055 
95194 



10, 



95333 
95473 
95613 

95753 
95894 



JD 



10 



,96035 
.96176 
.963I8 
, 9646 1 
, 96603 



10 



96746 
,96889 

97033 
97177 

97322 



10 



.97467 
,97612 

.97758 
,97904 
,98056 



10 



.9819? 

.98345 
,98492 

, 98646 
.98789 



10 



98938 
.9908^ 

.9923^ 
.9938^ 

•99538 



10 



10 
1 1 



99689 
,99841 

99993 
,00145 
,00293 



II 



00451 
00605 
00759 
00914 
01069 



II 



01225 
01381 

0153^ 
01694 
01852 



1 1 .02010 



Log. Kxsen. 



34 
35 
35 
35 

35 

36 

36 
37 
37 

Zl 
3l 
38 
38 
39 

39 
39 
40 
40 
40 

41 
41 
42 

42 
42 

43 
43 
44 
44 
44 

45 

45 

45 
46 

46 
47 
4^ 
4? 
48 

49 
49 
49 
50 
50 
51 
51 
51 
52 
52 
53 

53 
54 
54 
55 
55 

55 
56 
56 
57 
S7 
58 



7> 



Log. Vers. 



9.96039 
96053 
96067 
96081 
96095 



96 log 

96122 
96136 
96150 
96163 



9617^ 
9619I 
96205 

962 1 8 
96232 



96246 
96259 
96273 
96287 
96301 



96314 

96328 
96342 

96355 
96369 



96383 
96397 
96416 
96424 
96438 



1> Log 



96451 
96465 
96479 
96492 
96506 



96519 

96533 
96547 
96566 

96574 



96588 
9660T 
96615 
96629 
96642 



96656 
96669 
96683 
96697 
96716 



96724 

9673^ 
96751 
96764 
96778 



96792 
96805 
96819 
96832 
96846 



9.96859 



I I 



1 1 



II 



1 1 



I I 



I I 



II 



II 



II 



1 1 



II 



II 



I I 



Lotf. Vers. 7> |L<»er 

436 



Exsec. 



02010 
02163 
0232^ 
02487 
02646 



02807 
02968 
03129 
03291 
03453 



036 1 6 
03780 

03944 
04108 

04273 



04438 
04604 

04771 

04938 
05106 



1) 



05274 

05443 
05612 
05782 
05952 



06123 
06295 
06467 
06640 
06813 



06987 
07 161 

07336 
07512 

07688 



07865 
08043 
08221 
08400 
08579 



08759 
08940 

09121 

09303 
09486 



09669 

09853 
10038I 
10223I 
10409 



10595 
10783 
1097 1 
1 1 160 

1 1 349 



1 1 539 
1 1 736 
11922 
12114 
1230^ 



12501 



Kxs«>«' 



58 
59 
59 
59 
66 
61 
61 
61 
62 

63 
63 
64 
64 

65 

65 
66 

67 

67 

67 

68 

69 
69 

69 
70 

71 
71 
72 
73 
73 
74 
74 
75 
76 
76 

77 
71 
78 
79 
79 
80 
86 
81 
82 
82 

83 
84 
85 
85 
86 

86 
87 
88 

89 
89 

90 
91 
91 
92 

93 

93 



5 
6 

7 
8 

10 

1 1 
12 

13 

14 

15 
16 

17 

18 

19 



20 

21 

22 

23 

24 



35 
36 
37 
38 
39 



40 

41 

42 

43 
44 



45 
46 
47 
48 
49 



50 

51 
52 

53 
54 



55 
56 
57 
58 
59 



(>0 



J) 



p. p. 



6 


190 

19.0 


7 
8 


22.1 

25-3 


9 


28.5 


10 
20 


63-3 


30 


95 -o 


40 

50 


126.6 
158.3 



170 

17.0 

19-8 
22.6 

25-5 

28.3 

56.6 
85.0 

"3-3 
141-6 



180 

18.0 
21.0 
24.0 
27.0 
30.0 
60.0 
00. o 
120.0 
150.0 



160 

16.0 
18. 6 
21.3 
24 .0 
26.6 

53-3 
80.0 

106.6 

^33-3 



150 140 



15.0 

17-5 
20.0 
22.5 
25.0 
50,0 
75-0 

lOO.O 
I2S.O 



14.0 
16.3 

18.6 

21 .0 

23 -3 

46.5 

70.0 

93-3 
116.6 





130 


9 


b 


6 


13.0 


0.9 


0. 


7 


15-1 


1 





0. 


8 


17-3 


I 


2 


I . 


9 


19-5 


I 


3 


1. 


10 


21.6 


I 


5 


I. 


20 


43-3 


3 





2. 


.30 


65.0 


4 


5 


4- 


40 


86.6 


60 


5- 


5« 


108.3 


7 


5 


6. 



6 


7 

0.7 


6 

0.6 


7 
8 


0.8 
0.9 


0.7 
0.8 


9 


I.O 


0.9 


10 


I.I 


1 .0 


2D 


2.3 


2.0 


30 


3-5 


3-0 


40 


4-6 


4.0 


50 


5-8 


5-0 



5 

05 
0.6 

0.6 
0.7 

0-8 
1-6 

2-5 

3-3 
4.1 





14 


14 


I 


6 


1.4 


1.4 


I. 


7 


1-7 


1-6 


I. 


8 


1.9 


1-8 


I. 


9 


2.2 


2.1 


2. 


10 


2.4 


2-3 


2. 


20 


4-8 


4 6 


4- 


30 


7.2 


7.0 


6. 


40 


9-6 


9-3 


9- 


50 


12. 1 


11.6 


II. 



r. »' 



TABLE VIII. -LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 

80° S7'^ 







10 

II 

12 

14 



15 
i6 

I? 
i8 

19 



20 

21 

22 

23 
24 



^5 
26 
27 



29 



30 

31 
32 
33 
J±. 
35 
36 
37 
38 
39 
40 

41 

42 

43 

44 

45 
46 

47 
4B 
49 



50 

51 

52 
53 
54 



55 
56 
57 
58 

5?_ 
GO 



Loar. Vers. 



J) 



9.96859 
.96873 
.96887 
.96900 
.96914 



,96927 
,96941 
96954 
96968 
96981 



9.96995 
.970O8 
. 97022 
•9703? 
• 97049 



.97062 

.97076 

.97089 

97103 

97116 



■ 97 1 30 
■97143 
■97157 
,97170 

97183 



■97^97 
.97216 
,97224 
,9723^ 
97251 



,97264 
,97277 
,97291 

97304 
97318 



997331 
•97345 
•97358 
•97371 
•97385 



9^97398 
.97412 

•97425 
•97438 
•97452 



9.9746S 

•97478 
.97492 

•97505 
•97519 



9-97532 
•97545 
•97559 
•97572 

•97585 



9^97599 
.97612 
.97625 

•97639 
.97652 



9.9766: 



13 
14 
13 
13 

13 
J3 
f3 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 



13 

13 
13 
13 
13 
13 
13 
J3 
13 
13 



liOp. Kxsec. 7> 



I I 



. 12501 
. 12696 
. I2891 
.13087 
.13284 



I I . 1^482 
. I 3680 

•13879 
.14079 
. 14286 



'95 
195 
'96 
'96 
198 
198 
199 
200 
201 



1 1 



14482 
14684 
14887 
15092 
15297 



1 1 



15502 
15709 
1-5917 
16125 
'6334 



1 1 



16544 
16755 
16967 
17186 
17394 



1 1 . 1 7609 
. 17824 
. 1 804 1 
.18259 
.1847^ 



II . 18697 
.1891^ 

•i9'38 
• '9361 
.19584 



II 



. 19809 
. 20034 
.20261 
. 20489I 

.20717, 



11.20947, 
.211781 
.21410I 
.21643' 
.21877: 



I I .221 I2j 
.22349 
.22586 
.22825 
.230651 



' I • 233O6 

•23548 
. 23792' 

-240371 
.24283! 



11.24530 

•24778 
•25028 
.25279 
•255311 



11.25785 



201 
202 
203 
204 
205 

205 
206 
208 
208 
209 

210 
21 1 
212 

213 
214 

214 
215 

216 

218 

218 

219 
226 

221 
222 
223 
224 
225 
227 
227 
228 
230 
236 
232 

233 
234 

235 

236 
237 
239 
239 
24? 
242 

243 
245 
246 

247 

248 
250 

251 
252 
254 



Lojf. Vers. I It 



9.97665 
.97679 
.97692 
•97705 
•977I8 



9.97732 

•97745 
•97758 
.97772 

■97785 



9^97798 
.97811 

.97825 

•97838 

•97851 



9.97864 
•97878 
.97891 

• 97904 
-97917 



9-97931 
•97944 
•97957 
.97976 
.97984 



9-97997 
.98016 
.98023 

• 98036 
.98050 



9.98063 
.98076 
. 98089 
.98102 
.98116 



9.98129 
.98142 

•98155 
.98168 
.98181 



9.98195 
.98208 
.98221 

•98234 
•9824? 



9.98266 

•98273 
.98287 
.98300 

•98313 



9.98326 

•98339 
.98352 

.98365 

•98378 



9.98392 

• 98405 
.98418 
.98431 
•984-14 



f). 984^7 



Loir. Vers. | I> \\,nK. Kxser. /> k Loir. Vers. 



'3 

13 
'3 
13 

'3 
13 



13 



13 
'3 
'3 
'3 

13 
13 
13 
13 
'3 

13 
13 
'3 
13 
13 

13 
13 
13 
13 
13 

13 
'3 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
'3 
'3 
13 
13 
13 
13 

M 



13 
13 

'3 
'3 



!-(»::. K\> 



/> 



"•25785 _ 

.26046 "^5 

.26297 "^0 

•26554 ^'^ 
.26814 



1 1 



27074 

27336 
27599 

27864 
28131, 



1 1 



•28398 
.28668 
.28938 
.2921 1 

•29485 



1 1 



.29766 
• 3003^ 
.30316 
•30596 
•30878 



1 1 



^,1 162 
•3'447 
•31734' 
.32023 

•32313 



II 



.32606 
. 32900 
•33196 
•33494 
•33793 



1 1 . 34095 

■34398 
• 34704 
.35011 

•35321 ^ 

''•356321 ^I^ 



-5/ 
259 

266 

262 

263 

265 

266 

267 

269 

276 

272 

274 

275 

277 
278 
279 
282 
2S3 
285 
287 

288 
296 

292 

294 
296 
298 
299 

301 
303 
305 
307 
309 



10 

1 1 

12 

'3 
14 

'5 
16 

17 
18 

'9 



I'. I'. 



35946 
.36261 

•36579 
• 36899 



1 1 



.37221 

•37546 
.37872 
.38201 
•38532 



1 1 . 38866 
.3920I 
•39540 
•39886 
.40224 



1 1 .40569 
. 409 1 8 
.41269 
.41622 
•41979 



7> 



11.42338 
.42699 
.43064 

•4343' 
.43802 

"-44175 



3^3 
3'5 
318 
320 
-7 2 '> 

324 
326 
328 
33' 

333 
335 
338 
340 

343 

345 
348 
351 
353 
356 

359 
361 

36.1 

37 5 



20 

21 

22 

23 

3_ 

25 

26 

27 
28 

29 



;iO 

31 
32 
33 

ii 
35 
36 

37 
3^ 
39 



j> 



40 

41 
42 
43 

J4 

45 
46 

47 
48 

49 
50 

51 
52 

53 

55 
56 
57 
58 
59 
(;o 



6 


250 

25.0 


7 
8 


29.1 
33-3 


9 


37-5 


10 
20 


83.3 


30 
40 
50 


125.0 
166.^ 
208.3 



230 

23.0 

26. § 

30 6 
34-5 
38^3 
76.6 
115. o 

'53^3 
191. 6 





210 


6 


21 .0 


7 


24^5 


8 


28.0 


9 


3i^5 


10 


35^o 


20 


70.0 


20 


105.0 


40 


140.0 


50 


»75-o 



240 

24 .0 

28.0 

32 o 
36.0 
40.0 

80.0 

120.0 
160.0 

200.0 



220 

22.0 
25-6 
29.3 
33^o 
36.^ 

73-3 
1 10. o 

'83.3 



200 

20.0 
2^-3 
26.6 
30.0 

33-3 

66.6 

100.0 

133.3 

166.6 



0.1 
0.1 



0-3 
0.5 
o.^ 





14 


13 


6 


1.4 


'•3 


7 


'•6 


i.ti 


8 


1-8 


1.8 


9 


2.1 


2.0 


10 


2-3 


3.2 


20 


4.6 


4..S 


30 


7.0 


6.7 


40 


9.3 


9.0 


50 


".6 


II. 2 



0.1 
o. t 



13 

'•3 

'•7 
1.5 
2. 1 
4-3 
6..S 
8.6 
10. 8 



r. r. 





190 


4 


3 


6 


ig.o 


0.4 


o^3 


7 
8 

9 


22. 1 

25^3 
28.5 


0.4 

0-5 
0.6 


0-3 
0.4 
0.4 


10 
20 


3'-6 
633 


°'6 
^•3 


0-5 
I.O 


30 
40 
50 


95-0 
126.6 
15S.3 


2.0 
3.3 


'•5 
2.0 

= •5 1 



437 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
88° 89" 



' Lo 


?. Vers. 


J> Loi:. 


Exsec. 1 J> Loj 


J. Vers. 


I) 


Loff. Exsec. 


2> 


/ 


P. P. 


9 


9845^ 


I^ " 


44175] 376 9 

44551! :^7a 


99235 




11.75050 


742 

755 
768 
781 







I 


98476 


13 


99248 


1 Z 


.75792 


I 




2 

3 
4 


98483 

98496 

.98509 


13 

13 
13 


44931 ^g 

45313 ^86 
45699 ^., 


99261 
99274 
99287 


13 
13 
13 


.7654^ 
.77316 
.78097 


2 

3 
4 




5 9 


.98522 


\l '' 


46088 ^^^ 9 


99299 


12 


11.78892 


795 
809 

825 
840 
856 

872 
896 
908 
927 


S 




6 


.98535 


13 


46486 ^92 
•46876 395 


99312 


13 


.79702 


6 




7 


•98548 


13 


99325 


13 


.80527 


7 




8 


.98562 


^3 


•47275 ii^ 


99338 


1 z 


.81367 


8 




9 


•98575 


13 


•47677 -^°; 


99351 


13 


.82223 


9 




10 9 


.98588 


;^ '■ 


48083 4-^ 9 


99363 


12 


11.83095 


10 




II 


.98601 


ij> 


48493 jj^ 


99376 


13 


.83986 


II 




12 


98614 


13 


•48906 4 3 


99389 


13 


. 84894 


12 




13 


.98627 


^j 


•49323 l^Q 


• 99402 




.85821 


13 




14 


98640 


13 


49743 "^^ 


•99415 


13 

13 
12 


.86768 


947 

967 
989 


14 

15 
16 




15 9 
i6 


98653 
98666 


\l ■' 


50168 ;|^| 9 

50597 f:^ 


99428 
99446 


11.87735 
.88724 


17 


98679 


13 


51029 2:^: 


99453 


13 


•89735 




17 




i8 
19 


98692 
98705 


13 
13 


5H66 436 
51906 ^^^ 


. 99466 
•99479 


1 2 

13 


. 90769 
.91829 


1 1034 
1059 

1085 
1112 
1 146 
1171 
1203 

1236 


18 
19 




20 9 

21 


98718 
98731 


\i " 


52351 445 g 

53713 % 
54176 4 3 


•99491 
•99504 


12 
13 


I 1 .92914 
.94026 


20 

21 




o -7 


98744 


13 
13 
13 


•99517 


i^ 


•95167 


22 




23 
24 


98757 
98770 


•99530 
•99543 


13 
12 


•96338 
•97541 


23 
24 

25 




25 9 


98783 


M " 


54643!^? 9 


99555 


11.98777 


26 


98796 


13 


•99568 


^3 


I 2 . 00048 


' \ 


26 




27 


98809 


13 


5|5^l 485 
56076 4 

56563 "^^l 


99581 




.01358 


1309 


27 




28 
29 


98822 
98835 


13 


99594 
99606 


13 
12 

13 

T 


.02707 
• 04098 


1349 
1 391 

1436 
1485 


28 
29 

30 




30 9 


98848 


=3 ": 


57334 roi 


99619 


12.05535 


31 


98861 


99632 




.07020 


31 




32 


98874 


13 


58058 504 

58567; 1^:1 

59082 513 


99645 


3 


•08557 


1537 


32 




33 
34 


98887 
98900 


13 


99657 
99670 


13 


. IOI49 
. II80I 


1592 
1652 

1716 


33 
34 




35 9 


98913 


;^ II 


^ -^ ?20 

59602 J 9 

60129 527 

60662 533 


99683 


I 2 


I2.I3517 


35 




36 


98925 




99695 




• 15302 :^^f 


36 




37 


98938 


13 


99708 


13 


.17163 




37 




38 
39 


98951 
98964 


I J 
13 


61202 539 

61747. 545 


99721 
99734 


13 
12 

13 
12 


.19106 
.21139 


1943 
2033 

2I3I 
2246 
2361 


38 
39 
10 

41 
42 




10 9 

41 
42 


98977 
98996 
99003 


13 


62300 ^52 Q 


99746 
99759 
99772 


12.23271 
.25511 
.27872 


6 
7 


I 

I 
I 


3 

• 3 

.6 


13 

^•3 
1.5 


43 
44 


990 1 6 
99029 


13 
12 


S5g: . 


99784 
99797 


13 


.3036^ 
•33013 


2495 
2645 
2815 


43 
44 


9 

lO 


2 
2 



2 


1-7 

2.1 


45 9 


99042 


'5 II 


65167 


soo 


99810 


12 


12.35828 


45 


30 


4 
6 


•7 


4-3 

6.5 


46 


99055 


ij 


65762 


99823 


3 


•3883? 


3009 


46 


40 


9 


.0 


8.6 


47 


99068 


13 


66366 ^Vl 
f 978 ^4 
67598 . . 


99835 


1 2 


. 42068 


3231 
3489 
3791 
4152 

4588 
512^ 
5812 
6707 


47 


50 


1 1 . J 1 


10. § 


48 
49 


99081 
99093 


13 
12 


99848 
99861 


13 
12 
12 

13 
12 
12 


•4555^ 
•49349 


48 
49 
50 

51 

52 
53 
54 


1 


50 9 

51 
52 
53 
54 


99106 
991 19 
99132 

99H5 
99158 


13 
13 
12 


68227l^;8 9 

68865I ^38 

695ii'6t^ 
70168 ^56 

70834: f' 


99873 
99886 

99899 
9991 1 

99924 


12.53501 
. 58089 

.63217 
.69029 

•75736 


6 

7 
8 

9 


] 


[2 

1.2 
1.4 

1-6 
1.9 


55 9 

56 

57 

58 

59 


99171 
99184 
99197 
99209 
99222 


13 
12 

13 
1 1 


7^509J68| 9 

728921 96 
736001 / / 
74319' II 


99937 

99949 
99962 

99974 
99987 


13 
12 

12 
12 

13 
12 


12.83667 

•93371 
13.05877 

•234991 
•53615 


7931 

9704 

I25O6 

I762T 

301 16 


55 
56 
57 
58 
59 
00 


10 
20 
30 
40 
50 


I 


2. 1 
4.1 

6.2 

8.3 
0.4 


(>0 9 


99235 


^ II 


75050 10 


00000 


Infinity 


1 


Lo 


u. Vers. 


D Log. 


Kxsec' T> Lo! 


:. Vers. 


j> 


Loar. Exsec. 


/> ' 1 


P. P. 1 



438 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 





lO 

20 

30 
40 

50 

1 

10 
20 

30 
40 

50 

2 

10 
20 

30 
40 

50 

3 

10 
20 

30 
40 

50 

4 

10 

20 

30 
40 

50 

5 
10 

20 

30 

40 

50 

e 

10 

20 

30 
40 

7 

10 

20 

30 
40 

50 

8 

10 

20 

30 
40 

50 

9 

10 
20 

30 
40 

50 
10 



Sin. 



0.0000 



0.0029 
0.0058 
o.ooSf 
o.ci 15 
0.0I4I 



0.0174 



0.0203 
0.0232 
0.0262 
0.0291 
0.0320 



0.0349 



0.0378 

0.0407 

0.0436 

0.0465 

0.0494 



0.0523 



0.0552 

0.0581 
0.0610 
0.0639 
0.0663 



0.0697 



0.0726 
0.0755 

0.0784 
0.0813 
0.0842 



0.0871 



o. 0900 
0.0929 
0.0958 
0.0987 
o. 1016 



0.1045 



0.1074 
O.I 103 
O.I 1 32 
0.II6I 
o. 1 190 



0.I2I8 



o. 1 247 
O.I276 

0.1305 
0.1334 
0.1363 



0.I39I 



0.1420 

0.1449 
0.1478 
o. 1 507 

0.1535 



0.1564 



o. 1 593 
0.1622 
0.1656 
0.1679 
0.1708 



0.1736 



Cos. 



29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

28 

29 
29 

29 
29 

28 

29 
29 
29 

28 

29 

28 

29 
29 

28 

29 

28 

29 

28 

29 

28 
2§ 

29 



d. 



Tan. 



J). 0000 

0.0029 
0.0058 
0.0087 

o.oi 16 
0.014.5 
0.0174 



0.0203 
0.0233 
0.0262 
0.0291 
0.0320 



d. 



0.0349 



0.0378 
o. 040^ 
0.0436 
0.0466 
0.0495 



0.0524 

0-0553 
0.0582 
0.061 1 
0.0641 
0.0670 



0.0699 

0.0728 
0.0758 
0.0787 
0.0816 
0.0845 



0.0875 



o. 0904 

0.0933 

0.0963 
0.0992 
o. 102T 



0.I05I 



o. 1 080 
o. mo 

0.1139 
o. 1169 

0.1 198 



0.1228 



0.1257 
0.1287 

0.1316 
0.1346 
0.1376 



o.i4og 



0.1435 
0.1465 
0.1494 
o. 1524 
0.1554 



0.1584 



0.1613 
0.1643 
0.1673 
0.1703 

0.1733 



0.1763 



Cot. 



Cot. 



00 



343.773 
171.885 

114.588 
85. 9398 
68.7501 
;57_^89§ 
49.1039 
42.9641 
38.1884 

34-367^ 
31.2416 



28.6362 

26.4316 

24. 54 if 
22.903^ 
21.4704 
20.2055 



19.0811 



18.0750 
17. 1693 

16.3498 
15.6048 

14.9244 



14.3005 



13.726^ 
13.1969 
12.7062 
12.2505 
11.8261 



11.4300 



11.0594 
10.71 19 
10.3854 
10.0786 
9.788T 



9.5143 



9-2553 
9.0098 

8.7769 

8.5555 
8.3449 



8.1443 



7-9530 
7.7703 

7-595? 
7.4287 

7.268^ 



7- 1 153 



6.9682 
6.8269 
6.6911 
6.5605 
_6^3j+8 
^,3i3'7. 
6. 1976 
6.0844 

5-9757 
5.8708 

5-7693 



d. 



5-6713 



i860 
1398 

7756 
8217 
1261 
6053 
2046 



Tan. 



6380 

4333 
2648 
1244 
0061 
9056 
8'95 
7450 
6804 
6237 

5739 
5298 
4907 

4557 
4243 
3961 
3706 
3475 
3265 

3073 
2899 

2738 
2590 
2454 
2329 
2213 
2106 
2006 

1913 
1826 
1746 
1670 
1599 
1534 
1471 
1413 
1358 
1306 

1257 
1211 

1 1 67 
1126 
1087 
1049 
1014 
986 

<1. 



Cos. 



1. 000 

1. 0000 

I.GOOO 

0.9999 

0-9999 
0.9999 

^9998 

0.9998 
0.999? 
0-9996 
0.9996 
0.9995 



0-9994 

0.9993 
0.9991 
0.9996 
0.9989 
0.9988 



o.998g 



0.9984 
0.9983 
0.9981 

0.5979 
0.9977 



0997g 



0.9973 
0.9971 
0.9969 
0.9967 
0.9964 



0.9962 



0.9959 

0-9956 
0.9954 

0.9951 

0.9948 

0.9945 



0.9942 
0.9939 

0.9935 
0.9932 
0.9929 



0992g 

0.9922 
0.9918 
0.9914 
0.9916 
^.9906 
0.9902 
0.9898 
0.9894 
0.9890 
0.9886 
0.9881 



J°^9l77: 

0.9872 

0.9867 

0.9863 

0.9858 

o.98y_ 

0.9848 

Sin. 



d. 



yo 

50 

40 

30 
20 
10 
so 

50 
40 

30 

20 
10 
88 

50 
40 

30 
20 

10 

87 
50 
40 
30 
20 
10 
86 

50 
40 
30 
20 
10 

85 

50 

40 

30 
20 
10 
84 

50 
40 

30 
20 
10 
83 

50 
40 

30 
20 
10 
82 

50 
40 



30 


4 

5 


20 


6 


10 


7 


81 


8 

Q 


50 




40 




30 




20 





10 
80 



r. I' 



30 29 29 



3-0 
6.0 
9.0 

12.0 
15.0 
18.0 

21 .0 
24.0 
27.0 



14.7 
17.7 

20-6 
23.6 
26.5 



2.9 

5-8 
8.7 

11.6 

M-5 
17.4 

20.3 
23.2 
26.1 



28 5 4 4 



2-8 


05 


0.4 


5-7 
8.5 


I.O 

1-5 


0.9 
1.3 


11.4 


2.0 


1.8 


14.2 


2.5 


2.2 


17. 1 


3.0 


2.7 


19.9 


3-5 


31 


22.8 


4.0 


3-t 


25.6 


4-5 


4.0 



0.4 



1.6 



3.2 
3-6 



3322 



4 
5 
62 

72 

82 
9I3 



3 0.3 

70.6 

0.9 

1.2 

1.5 



2.1 

2.4 

I 2.7 



1.5 

1.7 
2.0 



1-4 
1.6 
1.8 



0.4 

0.5 
0.6 



.ojo.7 

.20.8 
.310.9 



0.0 
0.1 



I X O 

0.1 

0.3 
0.4 

0.6 
0.7 
0.9 



0.2 
0.3 

0.3 
0.4 
0.4 



P. P. 



80-90 



439 



TABLE IX.— NATURAL SINES, 



COSINES, TANGENTS, AND COTANGENTS. 
10-20° 



/ 


Sin. 


d. 

28 


Tan. 


d. 


Cot. 


d. 


Cos. 


d: 




p. p. 


10 

10 


0.1736 


0.1763 


30 


5-6713 


949 


0.9848 


5 


80 

50 




0.1765 


0.1793 


5-5764 


0.9843 


20 


0.1793 


^8 


0.1823 


i^ 


5.4845 


919 
890 
862 
8^A 


0.9838 


b 


40 




30 
40 


0.1822 
O.1851 


29 

28 


0.1853 
0.1883 


30 
30 
30 
30 

• 

30 


5-3955 
5-3093 


0.9832 
0.9827 


5 
5 


30 
20 


33 32 31 


50 

11 

10 


0.1879 


28 
28 


O.1913 


5-2256 


811 
787 
764 
742 


0.9822 


5 
6 


10 
79 

50 


I 
2 
3 


3-3 3 
6.6 6 

9-9 9 


.2 3.1 
.4 6.2 

.6 9.3 


0.1908 


0.1944 

0.1974 


5-I44S 


0.9816 


01 936 


5.0658 


0.9816 


20 


0.1965 


^8 

28 


0. 2004 


3^ 
30 


4.9894 


0.9805 


5 
6 


40 


4 
5 


13.2 12 
16.5 16 


.8 12.4 
•0 15.5 


30 


0.1993 


2R 


0.2034 


30 


4.9151 


721 


0.9799 


5 


30 


6 


19.8 19 


.2 18.6 


40 
50 

12 

10 

20 


0.2022 
0.2050 


28 
28 
28 

28 


0.2065 
0.2095 


30 

30 

.30 

30 


4.8430 

4-7728 


7o£ 
682 
664 
646 
620 


0.9793 
0.978^ 


6 
6 
6 

6 
6 


20 
10 
78 

50 
40 


7 
8 

9 


23.1 22 
26.4 25 
29.7 28 


.4 21-7 
.6 24.8 
.827.9 


0.2079 


0.2125 


47046 


0.9781 

0.9775 
0.9769 


0.2I0f 
0.2136 


0,2156 
0.2185 


4.6382 
4-5736 




30 
40 


0. 2 1 64 
0.2193 


28 


0.2217 
0.224^ 


30 


4.5107 
4.4494 


613 


0.9763 
0.97 56 


6 


30 
20 


1 


36 30 29 1 

3.0 3.0I 2.9 ( 


50 
13 

10 

20 


0.2221 


28 

28 
28 

28 


0.2278 


3^ 
30 

31 
30 


4-3897 


597 
582 
568 
553 


0.9750 


6 
6 
6 
6 


10 

77 

50 
40 


2 

3 

4 
5 
6 


6.1 6 
9.1 g 

12.2 12 

15-2 15 

18.3 18 


.0 5.8 ! 
.0 8.7 

.0 II. 6 
.0 14.5 
.0 17.4 


2249 


O.23O8 


43315 


09743 


0.2278 
0.2306 


0.2339 
0.2370 


4-2747 
4.2193 


0-9737 
0.9736 


30 
40 


0.2334 
0.2362 


28 

28 


0.2401 
O.243T 


3i 
30 


4-1653 
4.1125 


540 
527 
515 


0.9723 
0.9717 


y 

6 


30 
20 


7 
8 


21.3 21 

24-4 24 


.0 20.3 
.0 23.2 


50 


0.2391 


23 


0. 2462 


31 


4.0616 


0.Q710 




10 


9 


27.4 27 


.0 26.1 


U 

10 

20 


0.2419 


28 

28 


0.2493 


31 
30 
31 


4.0108 


491 
480 
469 


0.9703 


7 
7 

7 


76 

50 
40 


20 28 2*7 


0.2447 
0.2475 


0.2524 
0.2555 


3-9616 
3-9136 


0. 9696 
0.9688 


30 


0.2504 


^8 


0.2586 


31 


3.8667 


0.9681 


7 


30 


I 


2.§ 2 


.8 2.7 


40 


0.2532 


28 


0.2617 


31 


3.8208 


458 


0.9674 


7 


20 


2 


5.2 5 


.6 5.4 


50 

15 


0.2560 


28 

28 


0. 2648 


3i 
31 


3-7759 


449 
439 

42Q 


0. 9665 


7 

7 
8 


10 
75 


3 
4 


8.5 8 
II. 4 II 


.4 8.1 
.2 10.8 


0.2588 


0.2679 


3.7326 


0.9659 


10 


0.2615 


?8 


0.2716 


31 


3.6891 


420 


0.9651 


7 


50 


6 


17.1 16 


.8^16. 2 


20 
30 


0.2644 
0.2672 


28 
"8 


0.2742 
0.2773 


31 


3.6470 
3.6059 


411 
403 


0.9644 
0.9636 


7 
8 


40 
30 


7 
8 


19.9 19 
22.8 22 


.618.9 
.4 21.6 


40 


0.2700 


28 


0.2804 




3-565? 


■^94 


0.9628 


8 


20 


9 


25-525 


.2 24.3 


50 
16 

10 


0.2723 


28 
28 
?8 


0.2836 


31 
31 
31 


3.5261 


387 
379 

371 


0.9626 


8 
8 
8 


10 
74 

50 


10 8 


0.2758 


0.286^ 


3-4874 


0.9612 


0.2784 


0.2899 


3-4495 


0. 9604 


20 


0.2812 




0. 2930 




3-4123 




0-9596 


8 
8 


40 


3 


[ i.o 0. 


90.8 


30 
40 


0.2840 
0.2868 


■'■1 
28 


0.2962 
0.2994 


32 


3-3759 
3-3402 


357 


0.9588 
0.9580 


30 
20 




> 2.0 I. 
J 3.02. 


8 1.6 
72.4 


50 
17 

10 


0*2896 


28 

27 
28 


0.3025 


3i 

32 
31 


3-3052 


350 
343 
337 


0.9571 


8 
8 
8 


10 
78 

50 


- 


M-o 3- 

; 504- 
3 6.0 5. 


6 3.2 
5 4-0 
44.8 


2923 


0.3057 

0.3089 


32708 


09563 


0.2951 


3.2371 


0.9554 


20 


0.2979 


28 


0.31 21 


32 


3.2040 


331 


0.9546 


8 


40 




r 7.0 6. 
i 8.0 7. 


3 5-6 
2 6.4 


30 


0. 3007 


2y 


0.3153 


32 


3-1716 


324 


0.9537 


y 


30 


c 


) 9.0 8. 


I 7.2 


40 


0.3035 


28 


0.3185 


32 


3-1397 


319 


0.9528 


8 


20 




50 


0.3062 


2y 


0.3217 


32 


3.1084 


313 


0.9519 


y 


10 




18 


0.3090 


27 


0.3249 


32 


3.0777 


307 


0.9516 


y 


72 


A »v A v 


10 
20 


0.31 18 
0.3145 


27 


0.3281 
0.3313 


32 
32 


3-0475 
3-OI78 


302 
296 


0.950T 
0.9492 


9 
9 


50 

40 


1 c 

2 I 


/ / 

.70.7 < 

•5 1-4 




D.60.5 1 

1.2 1.0 


30 


0.3173 


27 


0.3346 


32 


2.9887 




0.9483 


9 


30 


32 


.22.1 


1.8 1.5 


40 


0.3200 


•I] 


0.3378 


32 


2.9606 


286 


0.9474 


y 


20 


4 3 


.0 2.8 : 


2.4 2.0 


50 
19 

10 


3228 


27 
27 
27 
27 


0.3411 


32 
32 
32 
32 


2.9319 


277 
272 
267 


0.9464 


y 
9 
9 

Q 


10 
71 

50 


53 
64 

7 5 
86 


-73-5 , 

.5 4-2 : 

.2 4.9 t. 

-0 5.6 < 


3-02.5 
?.6 3.0 

^23-5 
I..8 4.0 ' 


0.3255 


03443 


2.9042 


0.9455 


03283 


0.3476 


2.8770 


0.9445 


20 


0.3310 


27 


0-3508 




2.8502 


263 


0.9436 


5 


40 


96 


•76.3 . 


5.4 4-5 1 


30 


0-3338 




0-354I 




2.8239 




0.9425 




30 




40 


0-3365 


27 


0-3574 


33 


2.7980 




0.9415 




20 




50 


0.3393 


■^■7 
27 


0.3607 


ii 
32 


2.7725 


254 
250 


0.9407 


y 
10 


10 
70 




I20 


0.3420 


0.3639 


2-7475' 


0.9397 




Cos. 1 


d. 


Cot. 


d. 


TaB. I 


d. 


Sin. 1 


d. ' P. p. 1 



70°-80' 



440 



TABLE IX.— NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS. 



20 

lo 
20 

I 30 
I 40 

30 

21 

10 
20 

30 
40 

50 

22 

10 
20 

30 
40 

! 50 

23 

j 20 

I 30 
' 40 

24 

JO 

I 20 

I 30 
40 

50 

25 

10 
20 

30 
40 
50 

26 

10 

20 

30 
40 

50 

27 

10 
20 

30 
40 

50 

28 
10 

20 

30 
40 

50 

29 

10 
20 

30 
40 

50 

30 



Sin. 



d. 



o 3420 



0-3447 

0.3475 
0.3502 

0.3529 
0-3556 



035 83 

o. 36 1 I 
0.3638 
0.3665 
0.3692 
0.3719 



03746 



0.3773 
0.3800 
0.3827 

0-3853 
0.3880 



o 3907 



0.3934 
0.3961 
0.398^ 

0.4014 
o. 404 1 



0.406^ 

0.4094 
0.4120 

0.4147 
0.4173 

0.4200 



o 4226 



0.4252 

0.4279 
0.4305 

0.4331 
0.4357 



04383 



0.4410 

0.4436 

0.4462 

0.4488 
0.4514 



04540 



0.4566 

0.4591 

0.4617 

0.4643 

0.4669 



0.4694 



0.4720 

0.4746 

0.4771 
0.4797 

0.4822 



0.4848 



0.4873 

0.4899 

0.4924 
0.4949 

0.4975 



0.5000 

Cos. 



Tan. 



3 639 

0.3672 
0.3705 

0.3739 
0.3772 
0.3805 



03838 

0.3872 

0.3905 
0.3939 
0.3972 

o. 4005 



0.4040 



0.4074 

0.4108 
0.4142 

0.4176 

0.4216 



0.4244 



0.4279 

0.4313 
0.4348 

0.4383 

o.44if 



04452 



0.4487 
0.4522 

0.455? 
0.4592 

0.462^ 



0.4663 



0.4698 
0.4734 
0.4770 
0.4805 
0.484T 



0487^ 



0.4913 

0.4949 
0.4986 
0.5022 
0.5058 



0.5095 



0.5132 
0.5169 
0.5205 
0.5242 
0.5280 



05317 



0.5354 
0.5392 
0.5429 
0.546? 
0.5505 



0-5543 



0.5581 
0.5619 

0.565? 
o. 5696 

0.5735 



05773 
Cot. 



33 
33 
33 
33 
33 
33 
33 
33 
33 
34 
34 
33 
34 
34 
34 
34 
34 
34 
34 
34 
35 
34 
35 
34 
35 
35 
35 
35 
35 
35 
35 
36 
35 
36 
36 
36 
36 
36 
36 
36 
37 
36 
37 
36 
37 
37 
37 
37 
37 
37 
38 
37 
38 
38 
38 
38 
38 
39 
38 

(1. 



Cot. 



2-7475 



2.7228 
2.6985 
2.6746 
2.6511 
2.6279 



2.6051^ 

"2:7826 
2.5604 

2.5386 
2.517T 
2.4959 



24751 



2.4545 
2.4342 
2.4142 

2-3945 
2.3750 



2.3558 



2.3369 
2.3182 
2.2998 
2.2815 
2.263^ 



2.2466 



2.2285 
2.21 13 

2.1943 
2.1775 
2. 1609 



2. 1445 



2. 1283 
2.II23 
2.0965 
2.0809 
2.0655 



2.0503 



2.0352 

2.0204 

2.0057 

.9911 

.9768 



9626 



.9486 

•9347 
.9210 
.9074 
.8940 

r88oy 



.8676 
.8546 

.841? 
.8296 
^8165 
.8040 



-791? 

■119% 
.7675 

.7555 
.743? 



7320 

Tan. 



247 
245 
239 
235 
232 
238 
225 
221 
218 
215 

212 
208 
206 
203 
200 
197 
194 
192 
189 
187 
184 
182 
179 

177 

175 
172 
I70 
168 
166 
164 
162 
159 
158 
156 

^54 
152 
»5o 
M8 
147 
145 
143 
142 
140 
139 
137 
136 

134 
132 

131 
130 

I2§ 
127 

125 
124 
123 
122 
126 
119 
118 
117 



Cos. 



<I. 



0.9397 

0.9387 

0.9366 

0.9356 
0.9346 



0.9336 



0.9325 

0.9315 
0.9304 
0.9293 
0.9282 



0.9272 



0.9261 
0.9250 

0.9239 
0.922^ 

0.9216 



0.9205 



0.9193 
0.9182 
0.9170 
0.9159 
0.9147 



0.913S 



0.9123 
0.911T 
o. 9099 
0.9087 
0.9075 



0.9063 



0.9050 

0.9038 
0.9026 
0.9013 
0.9006 



0.8988 



0.8975 
0.8962 
o. 8949 

0.8936 
0.8923 



0.8910 



0.8897 
0.8883 
0.8870 

0.8856 
0.8843 



0.8829 



0.8816 
0.8802 
0.8788 
0.8774 

0.8766 



0.8746 

0.8732 

0.8718 

0.8703 

0.8689 
^8675^ 

o. 8666 

Sin. 



13 

12 

13 
12 

13 
13 
13 
13 
13 
13 
13 
IS- 
IS 

13 
13 
14 
14 
13 
14 
14 

M 
14 

14 
14 
I-? 

d. 



70 

50 
40 
30 
20 
10 
69 

50 
40 
30 
20 
10 

68 
50 
40 
30 
20 
10 

67 

50 
40 

30 
20 
10 
66 

50 

40 

30 
20 
10 

65 

50 
40 

30 
20 
10 
64 

50 
40 
30 
20 
10 
63 

50 

40 

30 
20 
10 
62 

50 
40 
30 
20 
10 
61 

50 
40 

30 
20 
10 
60 



P. P. 



Z% 35 34 33 



1 3-5 

2| 7.1 

3 »o-6 

4J14.2 

5,17-7 

6 21.3 

7 24.8 
8;28.4 

9131.9 



3-5 

7.0 

10.5 

14.0 

17-5 
21.0 

24-5 
28.0 

31.5 



3-4 
6.8 



3-3 
6.6 

9-9 



13.6,13.2 
17.0 16.5 
20.4 IQ.8 

I 
23.8 23.1 

27.2 26.4 

30.6[29.7 



2^ 27 26 25 



2.7 


2-7 


2.6 


5-5 
8.2 


5-4 
8.1 


5-2 

7.8 


II. 


10.8 


10.4 


13-7 
16.5 


13-5 
16.2 


13.0 
15.6 


19.2 


18. q 


18.2 


22.0 


21.6 


20.8 


24.7 


243 


23-4 



1.4 
2,9 


1.4 

2.8 


1-3 
2.6 


4-3 


4.2 


3-9 


5-8 

7-2 

8.7 


5-6 
7.0 
8.4 


5-2 

6.5 
7.8 


10. i 
II. 6 


9.8 
11.2 


9.1 
10.4 


13.0 


12.6 


II. 7 





39 


38 


37 


36 


I 


3-9 


3-8 


3-7 


3-6 


2 


7-8 


7-6 


7-4 


7-2 


3 


11.7 


II. 4 


10. 1 


10.8 


4 


15-6 


15.2 


14.8 


14.4 


5 


19-5 


19,0 


18.5 


18.0 


6 


23-4 


22.8 


22.2 


21.6 


7 


27-3 


26.6 


25-9 


25.2 


8 


31.2 


30-4 


29.6 


28.8 


9 


35-1 


34-2 


33-3 


32.4 



2-5 

5-0 
7-5 

irf.o 
12.5 
15-0 

17-5 
20.0 

5 



14 14 13 12 



2.4 
3-6 

4.8 
6.0* 
7-2 

8.4 

9.6 

10.8 



10 

i.o 
2.0 
3-0 

4.0 
5-0 

6.0 

7.0 
8.0 
9.0 



II II 



il I 



9 
10.3 



2.2 

3-3 

4.4 

5-5 
6.6 

7-7 
8.8 

9-9 



P. P. 



60-70 



441 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS. 

30-40° 



o / 
lO 

20 

30 
40 

50 

31 

10 

20 

30 
40 

50 

32 

10 
20 

30 

40 

50 

33 

10 

20 

30 
40 

50 

34 

10 
20 

30 
40 

50 
85 

10 
20 

30 
40 

50 

36 

10 
2.0 

30 
40 

50 

37 

10 
20 

30 

I 40 

; 50 

|3$ 

10 

20 

30 

40 

50 

39 
10 
20 

30 
40 

50 

40 



Sin. 



0.5000 



0.5025 
0.5056 
0.5075 
0.5106 
0.5125 



0.5150 



0.5175 
0.5200 

0.5225 

0.5250 

0.5274 



0.5299 



0.5324 

o. 5348 

0.5373 

0.539^ 
o. 5422 



0.5446 



0.5471 
0.5495 
0.5519 

0.5543 
o.:;568 



0.5592 



0.5616 
o. 5640 
o. 5664 

0.5688 
0.5712 



0.5736 



0.5759 
0.5783 

o. 5807 
o. 5836 

0.5854 



0.5878 



0.5901 

0.5925 
0.5948 
0.5971 
0.5995 



0.6018 



0.6041 
0.6064 
0.6087 
0.61 15 
0-6133 
o6i5g 



0.6179 
0.6202 
0,6225 
0.6248 
0.6276 



0.6293 



0.6316 
0.6338 
0.6361 
0.6383 
0.6405 



0.6428 
Cos. 



d. 



Tan. 



25 
25 
25 
25 
25 
25 

25 
24 

25 
25 
24 
24 
25 
24 
24 
24 
24 

24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 

24 
24 

23 
24 

2§ 
23 
24 

23 
23 
23 
23 
23 
23 
23 

23 
23 

23 
23 
23 
23 
23 
23 
22 

23 
22 
22 

23 
22 



0.5773 



0.5812 
0.5851 
0.5896 
0.5929 
0.5969 



O.6OO8 



d. 



o. 6048 
0.6088 
0.6128 
0.6168 
0.6208 



0.6248 



0.6289 
0.6330 
0.6376 
0.641T 
0.6453 



0.6494 



0.6535 
0.6577 
0.6619 
0.6661 
0.6703 



0.6745 



0.678^ 
0.6830 

0.6873 
0.6915 
0.6959 



0.7002 



0.7045 
o. 7089 

0.7133 

0.7177 

0.7221 



0.7265 



0.7310 

0.7354^ 
0.7399 
0.7444 

0.7490 



0.7535 



0.7581 

0.7627 

0.7673 
0.7719 

0.7766 



0.7813 



0.7860 
o. 7907 

0.795^ 

o. 8002 
0.8050 



0.8098 



0.8146 
0.8194 

0.8243 

0.8292 

0.8341 



0.8391 

Cot. 



d. 

39 
39 
39 
39 
39 
39 
40 
39 
40 
40 
40 
40 
40 
41 
40 

41 
41 
41 
41 
41 
42 
42 
42 
42 
42 
42 

43 
42 

43 
43 
43 
43 
44 
44 
44 
44 
44 
44 
45 
45 
45 
4§ 

4S 
46 

46 
46 

46 
47 
47 
47 
47 
47 
48 
48 
48 
4§ 
49 
49 
49 
49 

"dT 



Cot. 



1.7320 



1.7204 
1.7090 
1.6976 
1.6864 

1.6753 



1.6643 



1-6533 
1.6425 
I.63I8 
1. 6212 
1.610^ 



1.6003 



1.5900 

1.5798 
1.5697 

1-5596 
1-5497 



1-5398 



1.5301 
1.5204 
I-5108 
1-5013 
1-4919 



1.4825 



1-4733 
1. 4641 
1.4550 
1.446c 
1.4370 



1.4281 



1-4193 
1. 4106 

1. 4019 

1-3933 

1.3848 



1-3764 



1.3680 
1-3597 
I-35H 
1.3432 
1. 3351 



1.3270 



1. 3190 
1.3111 
1.3032 
1.2954 
1.2876 



d. 



1.2799 



1.2723 
1 . 2647 
1. 2571 
1.2497 
1.2422 



I 2349 



1.2276 
1.2203 
I.2131 
1.2059 
1. 1988 



Tan. 



99 
98 
97 
96 
96 

95 
94 
93 
92 
92 

91 

90 



87 

86 
86 

85 
84 
84 
83 
83 
81 
81 
80 
80 
79 
78 
78 
77 
77 

76 
76 

75 
74 
74 
73 
73 
73 
72 

71 
71 
70 

d. 



Cos. 



0.8660 



o. 8645 

0.8631 

0.86I6 
o.86oi 
0.8586 



d. 



08571 



0.8556 
0.8541 
0.8526 
O.851I 

o. 8496 



0.8486 



0.8465 

o. 8449 

0.8434 
0.8418 

0.8/|02 



0.8388 



0.8371 

0.8355 
0.8339 

0.8323 

0.8306 



0.8290 



0.8274 

0.825^ 

0.824T 
0.8225 
0.8208 



0.8I9I 



0.8175 
0.8158 

0.8I4I 
o. 8 1 24 
0.8107 



0.8090 



0.8073 
0.8056 
0.8038 
0.8021 
o. 8004 



0.7988 



0.7969 

0.7951 
0.7933 

0.7916 

0.7898 



0.7880 



0.7862 

0.7844 

0.7826 

0.7808 
0.7789 



0-7771 



0.7753 
0.7734 

0.7716 
0.769^ 

0.7679 



0.7666 

Sin. 



d. 



F. P. 



60 

50 
40 
30 
20 
10 
59 

50 
40 
30 
20 
10 
58 

50 
40 
30 
20 
10 
57 

50 
40 

30 
20 
10 
56 

50 
40 

30 
20 
10 

55 

50 
40 
30 
20 
10 
54 

50 
40 
30 
20 
10 
53 

50 
40 

30 
20 
10 
52 

50 
40 
30 
20 
10 
51 

50 
40 

30 
20 
10 
50 



49 49 48 47 46 



49 4-9 

9-9i 9-8 

I4-8I4-7 

19.8 19.6 
24.724.5 
29.7,29.4 

34 6 34-3 
39.639.2 
44.544.1 



4.8^ 4.7 4.6 

9.6 9.4 9.2 

I4.4'i4.i|i3.8 



19. 
24.0 

^28.8 



18.8 18.4 
23-523.0 

z8.2 27.6 



33.632.9 32.2 
38.437.6,36.8 
43.2|42.3!4i.4 



4S 45 44 43 42 



4-5 
9.1 

13-6 


4-5 
9.0 

13-5 


4.4 

8.8 

13.2 


4-3 

8.6 

12.9 


18.2 


18.0 


17.6 


17.2 


22.7 


22.5 


22.0 


21-5 


27-3 


27.0 


26.4 


25.8 


31.8 
36.4 
40.9 


31-5 
36.0 

40.5 


30.8 
35-2 
39-6 


30.1 
34-4 
38.7 



4.2 

8.4 

12.6 

16.8 

21 .0 

25.2 
29.4 

33-6 
37.8 



41 41 40 39 



4.1 

8.3 
12.4 

16.6 
20.7 
24.9 

29.0 
33-2 
37-3 



4-1 

8.2 

12.3 

16.4 
20.5 
24.6 

28.7 
32.8 
36-9 



4.0 
8.0 



16.0 

20.0 
24.0 

28.0 
32.0 
36.0 



3-9 

7-8 

11.7 

15.6 

19-5 
23-4 

27-3 
31.2 

35-1 



2% 25 24 23 

2-3 

4.6 
6.9 



2.5 


2.5 


2.4 


5-1 


5-0 


4.8 


7-6 


7-5 


7.2 


10.2 


10. 


9.6 


12.7 


12-5 


12.0 


15.3 


15.0 


14.4 


17.8 


17-5 


16.8 


20.4 


20.0 


19.2 


I22.9 


22.5 


21.6 



22 22 18 



2.2 


2.2 


1-8 


6.7 


4-4 
6.6 


3-7 

5-5 


9.0 
II. 2 


8.8 
u.o 


7-4 
9.2 


13-5 


13.2 


II. I 


15-7 
18.0 


15-4 
17.6 


12.9 
14.8 


20.2 


19.8 


16.6 





If 


17 


16 


15 


1 


1.7 


1-7 


1.6 


1-5 


2 


3-5 


3-4 


3-2 


30 


3 


5.2 


5.1 


4.8 


4-5 


4 


7.0 


6.8 


6.4 


6.0 


5 


8.7 


8.5 


8.0 


7-5 


6 10.5 


10.2 


9.6 


9.0 


7 12.2 


II. 9 


II. 2 


10.5 


8 14.0 


13.6 


12.8 


12.0 


9 


15-7 


15-3 


14.4 


13-5 



9.2 
".5 

1^.8 

16.1 
18.4 
20.7 



18 

I. a 
3.6 

5-4- 

7. a 

9.0 

10. a 

12.6 

14.4 
16.2 



14 

1-4 
2.9 

4-3 

6.8 
7.2 
8.7 



II. 6 
13.0 



P.P. 



50-60 



442 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 

40°-45° 



40 

lO 

20 

30 
40 

50 
410 

10 
20 

30 
40 

50 

42 

10 
20 

30 
40 

50 

43 

10 
20 

30 
40 

50 

44 

10 
20 

30 
40 

50 

45 



Sin. 



0.6428 

0.6450 
0.6472 

o. 6494 

O.65I6 

0-6538 



0.6566 

0.658! 
o. 6604 
0.6626 
0.6648 
0.6669 



0.6691 



0.6713 
0.6734 
0.6756 
0.6777 

0-6798 
0.6820 



(1. 



0.6841 
0.6862 
0.6883 
o. 6904 
0.692^ 



0.694^ 



0.6967 
0.6983 
o. 7009 
o. 7030 
0.7050 



0.7071 



Cos. 



21 
22 
21 

22 

21 
21 
21 
21 
21 
21 

21 
21 
21 
21 
21 
21 

21 

21 
20 
21 
20 
20 

d. 



Tan. 



d. 



08391 

o. 8446 
0.8496 

0.8541 

0.8591 
o. 8642 

0.8693 



0.8744 

0.8795 

0.8847 
0.8899 

0.8951 



0.9004 



0.9057 

0.9IIO 

0.9163 

0.9217 
0.9271 



09325 

0.9379 

0-9434 
0.9489 
0.9545 
0.9601 



0.9657 



0.9713 
0.9770 
0.9827 
0.9884 
0.9942 



1. 0000 



49 
50 
50 
50 
51 
51 

51 
51 
52 
51 
52 
52 

53 
53 
53 
53 
54 
54 

54 
55 
55 
5S 
56 
56 

56 
56 
57 

57 
57 
58 



Cot. 



d. 



Cot. 



d. 



191-7 



iS4f 
1777 
1708 
1640 
1571 



1503 



1436 
1369 

1303 
1237 
1171 



.1106 



.1041 
.0977 
.0913 
.0849 

.0723 



.0661 

.0599 

.0538 

•0476 
.0416 



0355 



.0295 
.0235 
.0176 
.0117 
.0058 



0000 



70 
70 
69 
68 
68 
68 

67 
67 
66 
66 

65 
65 

64 
64 
64 
63 
63 
^3 

62 
62 
61 
61 
66 
66 

65 
59 
59 
59 
58 
58 



Cos. 



0.7666 



Tan. 



d. 



0.764! 
0.7623 

o. 7604 

O.75S5 
0.7566 



0-7547 



0.7528 
0.7509 
0.7489 
0.7476 
0.7451 



0.7431 



0.7412 
0.7392 
0.7373 
0-7353 
0-7333 



^7313 

0.7293 
0.7273 
0.7253 
0.7233 
0.7213 



0.7193 



0.7173 

0.7153 
0.7132 
0.71 12 
0.7091 



0.7071 



Sin. 



19 

18 
19 
19 
19 
19 

19 
19 
19 
19 
19 
19 

19 

19 

19 
20 

19 
20 

20 
20 
20 
20 
20 
20 

26 
20 
26 
26 
26 
26 



50 

50 
40 
30 
20 
10 

49 

50 
40 

30 
20 
10 

48 

50 
40 
30 
20 
10 

47 

50 

40 

30 
20 
10 

40 

50 
40 

30 
20 
10 

45 



p. r. 



70 


22 


22 


21 


21 


7.0 


2.2 


2.2 


2.1 


2. 1 


14.0 

21 .0 


i:l 


4-4 
6.6 


4-3 
6.4- 


42 
6.3 


28.0 


9.0 


8.8 


8.6 


8.4 


42.0 


II .2 
13-5 


11 .0 
13.2 


10.7 
12.9 


10.5 
12.6 


49.0 
56.0 
63.0 


15-7 
18.0 
20.2 


15-4 
17.6 
19.8 


15.0 
17.2 
19-3 


14.7 
16.8 
18.9 



69 20 20 19 19 



I 


6.9 


2.0 


2.0 


1.9 


2 

3 


13S 
20.7 


4-1 
6.1 


4.0 
6.0 


3-9 
5§ 


4 


27.6 


8.2 


8.0 


7.8 


5 


34-5 


10.2 


10. 


9-7 


6 


41.4 


12.3 


12.0 


II. 7 


7 

8 

9 


48.3 
55-2 
62.1 


14-3 
16.4 
18.4 


14.0 
16.0 
18.0 


13-6 
15-6 
17-5 



5-7 

7.6 

9-5 
II. 4 

133 
15.2 
17. 1 



68 68 67 66 18 



68 


6.8 


6.7 


6.6 


13-7 
20.5 


13-6 
20.4 


134 
20.1 


13.2 
19.8 


27.4 
34-2 
41. 1 


27.2 

34-0 
40.8 


26.8 

33-5 
40.2 


26.4 

33 
39-6 


47-9 
54-8 
61.6 


47.6 

54.4 
61.2 


46.9 
60.3 


46.2 
52.8 
59-4 



3-7 
5-5 



7-4 
9.2 



12.9 
14.8 
16.6 



6S 64 64 63 62 61 66 59 59 58 58 5l 57 56 56 55 54 54 53 53 52 52 



I 


6.5 


6.4 


2 

3 


19-6 


12.9 
19-3 


4 


26.2 


25.8 


5 
6 


327 
39-3 


32.2 
38.7 


7 
8 

9 


45-8 
52.4 
58.9 


45-1 
5t.6 
58.6 



6.4 

12.8 

19.2 

25.6 
32.0 

38.4 

44-8 
51.2 
57-6 



6.3 
12.6 
18.9 

25.2 
31-5 
37-8 

44. 1 
50-4 
56.7 



12.4 
18.6 

24.8 
31.0 
37-2 

43.4 
49.6 
55-8 



6.1: 6.0 

12.3 1 2 . 1 

18.4 18. I 
i 

24.6 24.2 

30.7 30.2 
36.9 36-3 

43.0 42.3 
49.2 48.4 
55-31 54-4 



5-9 
II. 9 

17-8 

23.8 
29.7 
35-7 

41-6 

47.6 

53-5 



5-9 
II. 8 
17.7 

23.6 
29-5 
35-4 

41.3 
47.2 

53-1 



5-8 
II. 7 

17-5 

23.4 
29.2 

35-1 

40.9 
46.8 
52-6 



5-8 
II. 6 
17.4 

23.2 
29.0 
34.8 

40.6 
46.4 
52.2 



5-7 
"•5 
17.2 

23.0 
28.7 
34-5 

40.2 
46.0 
51.7 



5-7 
II. 4 
17. 1 

22.8 

28.5 
34-2 

39-9 
45-6 
51-3 



5-6 
"■3 
16.9 

22.6 
28.2 
33-9 

39-5 
45-2 

50-8 



5.6 
II .2 
16.8 

22.4 
28.0 
33.6 

39-2 
44.8 
50.4 



5-5 
II .0 
16.5 

22.0 
275 
33-0 

38.5 
44.0 

49-5 



5-4 
10. 
16.3 

21.8 
27.2 
32-7 

38.1 

43-6 
49.0 



5-4 
10.8 
16.2 

21 .6 
27.0 
32.4 

37-8 
43-2 
48.6 



5-3 
10.7 
16.0 

21.4 
26.7 



37-4 
42.8 
48.1 



5-3 5-2 
10.6 10.5 
159 15-7 



5-2 

10.4 

15.6 



21 .2 21 .0 20.8 
26.5 26.2 26.0 
31.8 31.5 31.2 



371 36.7 
42.4: 42.0 

47-7I47-2 



36-4 
41.6 
46.8 



S? 51 so 50 49 



5-i 
10.^ 


5-1 
10.2 


5-0 
10. 1 


50 
10. 


154 


15-3 


15.1 


15-0 


20.6 


20.4 


20.2 


20.0 


257 
30.9 


25 -5 
30.6 


25.2 
30.3 


25.0 
30.0 


36.0 
41.2 

46.3 


35-7 
40.8 

45-9 


35-3 
40.4 

45-4 


350 
40.0 
45.0 



4.9 
9.9 

I4§ 

19.8 
24.7 
29.7 

34-6 
39-6 
44-5 



Table for passing from Sexagesimal to Circular 
Measure. 



100 
200 
300 

40 
50 
60 

70 
80 
90 



Circular Meas, 



1.74 532 9 
3-49 065 8 
5.235988 

0.69 813 T 
0.87 266 4 
1.04 719 7 

1.22 173 6 
1.39 626 3 
1.570796 



' Cirt'ular Moas. " Circular Meas. 



10 
20 
30 
40 

50 
6 

7 
8 

9 



0.00 290 9 
0.00 581 8 
0.00 872 6 
o.oi 163 5 

0.0 1 454 4 

0.00 174 5 
0.00 203 6 
0.00 232 7 
0.00 261 8 



10 
20 
30 
40 

50 

6 

7 
8 

9 



45-50 



o. 00 004 8 
o. 00 009 7 
0.00 014 5 
0.00 019 4 

0.00 024 2 

0.00 002 9 
0.00 003 4 
0.00 003 9 
0.00 004 3 



443 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 

0°-10° 10°-20° 





lo 
20 

30 
40 

50 

1 

10 
20 

30 

40 

50 

2 

10 

20 

30 
40 
50 

3 

10 
20 

30 
40 

50 

4 

10 

20 

30 

40 

50 

5 

10 
20 

30 
40 

50 

6 

10 

20 

30 
40 

50 

7 

10 

20 

30 

40 

50 

8 
10 

20 

30 
40 

50 

9 

10 
20 

30 
40 

50 
10 



Ters. 



00000 



. 00000 
, 0000 I 
. 00004 
, 00007 
,00016 



.00015 



. 00020 
.00027 
.00034 
. 00042 
.00051 



00061 



. 0007 I 
.00083 
.00095 
.OOIOg 

.00122 



00137 



.00152 
.00169 
.00185 
. 00204 
.00223 



00243 



. 00264 
.00286 

■ 00308 
,00331 

■003Sg 



00386 



. 00405 
.00433 
. 00460 
. 00483 
.00518 



00548 



.00578 
.00616 
.00643 
.00676 
. 007 1 6 



00745 



.00781 
.00818 
.00855 
.00894 
.00933 



00973 



.01014 
.01056 
.01098 
.01142 
.01186 



.01231 



.01277 
.01324 
,01371 
,01420 
01469 



(1. 



10 
16 
II 
12 
13 
13 
15 

15 
16 
I? 
18 

19 
20 

21 
21 

22 

23 

24 
25 
26 

26 

2? 
28 

29 
30 

30 
32 



Exsec. d. 



.00000 



. 00000 
. 0000 I 
. 00004 
. 00007 
.00016 



.00015 



. 00020 
.00027 
.00034 
. 00042 
.00051 



.00061 



OI519 



Vers. 



33 
33 
35 
36 
36 
37 
38 
39 
40 

41 
42 

42 

43 

44 

45 
46 

47 
47 
48 
49 
50 

(1. 



. 0007 1 
.00083 
.00095 
.00108 
.00122 



00137 



.00153 
.00169 
.00187 
.00205 
.00224 



. 00244 

.00265 
.00285 
. 00309 
.00332 
•00357 



.00382 



. 00408 
•00435 
. 00462 
.00491 
.00526 



00551 



.00582 
.00614 
. 00647 
. 0068 I 
. 007 1 5 



00751 



.0078^ 
.00824 
.00863 
. 00902 
. 00942 



00983 



.01024 
.01067 
.01116 
.01155 
.01206 



0I24g 

.01293 

.01341 
•01396 

.01446 
.01491 



01542 

Exsec. 



TO 
16 

li 

12 

13 
14 
14 
16 
16 
17 
18 

19 
20 

21 
21 

22 

23 
24 
25 
26 
27 
2? 
28 
29 
30 

31 
32 
33 
34 
34 
35 

36 
37 
38 
39 
40 
41 

41 
42 
43 
44 

45 
46 

47 
48 

49 
50 
50 

^1 



10 

10 
20 
30 
40 
50 

11 
10 
20 

30 
40 
50 

12 

10 

20 

30 
40 

50 

13 

10 
20 
30 
40 
50 

14 
10 
20 
30 
40 
50 

15 
10 
20 
30 
40 
50 

IG 
10 
20 
30 
40 
50 

17 
10 
20 
30 
40 
50 

18 
10 
20 
30 
40 
50 

19 
10 
20 
30 
40 
50 

20 



Vers. I d. 



O1519 



.01570 
.01622 
.01674 
.01728 
.01782 



• oi83'7 



.01893 
.01950 
. 02007 
. 02066 
.02125 



,02185 



.02246 
.02308 
.02376 
.02434 
. 02498 



02563 



.02629 
.02695 
.02763 
.02831 
.02906 



02970 



•03041 

•03113 
.03185 

•03258 
•03332 



03407 



03483 

•03559 
.03637 

.03715 
•03794 



•03874 

•03954 
.04036 
.04118 
.04201 
.04285 



04369 



•04455 
.04541 

.04628 
.04716 
. 04805 



04894 



. 04984 
.05076 
.0516^ 
.05266 

•05354 



05448 

•05543 
.05639 
.05736 
•05833 
■05931 



51 

52 
52 
53 

54 
55 

55 

57 
5? 
58 

59 
60 

61 

62 
62 

63 
64 

65 
66 

66 
6^ 

68 
69 
70 

76 

72 
72 
72 
74 
75 

75 
76 
7? 
78 

79 
80 

80 
81 
82 

83 
84 
84 

85 
86 

8? 
8? 
89 
89 
90 
91 
91 
93 
93 
94 

95 
95 
97 
9? 
98 
99 



Exsec. 



06036 
Vers. I d. 



01542 



.01595 
.01648 
.01703 

•OI758 
.01814 



.01871 



.01929 
.01988 
. 02048 
.02109 
.02171 



, 02234 



.0229^ 
.02362 
.02428 
. 02494 
.02562 



02636 



.02700 
,02770 
,02841 
.02914 
.02987 



.03061 

•03136 
.03213 

.03290 

.03368 

•0344? 



03527 



. 03609 
.03691 
•03774 
•03858 
•03943 



04030 



.04117 
.04205 
.04295 

.04385 
.04475 



04569 



. 04662 

•0475^ 
.04853 
.04949 
.0504? 



05146 



.05246 
•05347 
•05449 
.05552 
.05655 



05762 



.05868 
.05976 
.06085 
. 06 I 94 
.06305 



■ 06418 

Exsec. 



52 

53 
54 
55 
56 
57 
58 

59 
60 

61 
62 
62 

63 
65 
65 
66 
6f 
68 
69 
70 
71 
72 
73 
74 

75 
76 

77 
78 

79 
80 

81 
82 

83 
84 
85 
86 
87 
88 

89 
96 

91 

92 

93 

95 

95 

96 

98 

98 

100 

loi 

102 

103 

104 

105 

105 
I of 
109 
109 
III 

112 



P. P. 



110 100 90 80 70 60 50 40 



II 


10 


9 


8 


7 


6 


5 


22 


20 


18 


16 


14 


12 


10 


33 


30 


27 


24 


21 


18 


15 


44 


40 


36 


32 


28 


24 


20 


55 


50 


45 


40 


3S 


30 


25 


66 


60 


54 


48 


42 


36 


30 


77 


70 


63 


56 


49 


42 


35 


88 


80 


72 


64 


56 


48 


40 


99 


90 


81 


72 


b3 


54 


45 



30 20 10 9 9 8 8 :7 



3 


2 


I 


0.9 


0.9 


0-8 


0.8 


6 


4 


2 


1.9 


1.8 


1-7 


1.6 


9 


6 


3 


2-8 


2.7 


2.5 


2.4 


12 


8 


4 


3^8 


3^6 


3-4 


3^2 


15 


10 


5 


4-7 


4^5 


4.2 


4.0 


18 


12 


6 


5-7 


5.4 


5-1 


4.8 


21 


»4 


7 


6-6 


6.3 


5.q 


5.6 


24 


16 


8 


7.6 


7.2 


6.8 


6.4 


27 


18 


9 


8.5 


8.1 


7-6 


7.2 



0.7 

1-5 
2.2 

3-0 

3-7 
4-5 

5-2 
6.0 
6.7 



7 6 6 5 5 4 4 



0.7 0.5 0.6 0.5 0.5 0.4 0.4 
1.4 1.311.2 I.I 1.00.9 0.8 

3.1 1.9 1.8 1. 6 1.5 1.3 1.2 

2.8 2.6 2.4 2.2 2.0 1.8 1.6 

3.5 3.2 3.0 2.7 2.5 2.2 2.0 

4.2 3.9 3.6 3.3 3.0 2.7 2.4 

4.9 4.5 4-2 3-8 3-5 3-1 2-8 

5.6 5.2 4.8 4.4 4.0 3.6 3.2 

6.3 5-8 5-4I4.9 4^5 4-03-6 



3 3 2 2 I I o 



03 

0-7 
1 .0 



2.4 

2.8 



3-1 



.2 

•5 
.8 

2.1 

•4 
2.7 



0.2 

0.5 
0.7 

1 .0 
1.2 
1.5 

1-7 
2.0 



0.4 
0.6 



0.3 

0.4 



0.6 
0.7 
0.9 

i.o 



o. 1 
0.2 
0-3 



0.4 

0-5 
0.6 



0.3 



70-3 

0.8 0.4 

9I0.4 



P. P. 



444 



TABLE X. — NATURAL VERSED SINES AND EXTERNAL SECANTS. 

2ir-:ur 'M) ur 



20 

lO 

20 

30 
40 

50 

21 

10 

20 

30 
40 

50 

22 

10 
20 

30 
40 

50 

23 

10 
20 

30 
40 

50 

24 

10 
20 

30 

40 

50 
2o 

10 
20 

30 
40 

50 
2G 

10 
20 

30 
40 

50 

27 

10 

20 

30 
40 

50 

28 
10 
20 

30 
40 

50 

29 

10 
20 

30 
40 

50 
80 



Vers. 



d. 



, 0603 

■ 0613^ 

.0623 

•0633 

.0643 

.0654 



.0664 



.0674 
.0685 
,0696 

07O6 
,0717 



0728 

0739 
,0750 

.0761 

.0772 

.0783 



,079 s 



.0S05 
,0818 
.0829 
.0841 
.0853 



0864 



,0876 
,0888 
,0906 
.0912 
.0024 



0937 



,0949 
.0961 
,0974 
,0986 
.0999 



I0I2 



1025 
103^ 
1050 
1063 
1077 



1090 



• I 103 

.1116 
.1130 

.1143 

• II 57 



1170 



.1184 
.1198 
. 1212 

. I22§ 
.1239 



^i25_4_ 

.1268 
.1282 
.1296 
• 131 1 
•1325 
1339 

Vers. 



10 
10 
10 
10 
id 
10 
16 
16 
II 
10 
II 
10 

II 
II 
II 
II 
I I 

iT 
II 
II 
II 
II 
12 
II 

12 
12 
12 
12 
12 
12 

12 
12 
12 
12 

13 

12 

13 
12 

13 
13 
13 
13 

13 

13 
13 
13 
13 
13 

13 
14 
14 
13 
14 
14 

14 

14 

I ^4 

' 14 



Kxsec. 



0642 



,0653 
,0664 
, 0676 
,0688 
.0699 
^711 



.0723 

.0735 
.0748 

.0760 

.0772 



078g 

.0798 
,0811 
.0824 
.0837 
.0850' 



,0863 



.0877 
.0890 
.0904 

.0918 
.0932 



.0948 



,0966 
,0975 
.0989 
. 1004 
. 1019 



1034 



1049 
1064 
1079 
1094 
1 1 16 



1126 



1 142 
1 1 58 
1174 
1196 
1206 



1223 



. 1240 
.1257 
.1274 
. 1291 
.1308 



1325 



•1343 
• 1361 
•1379 
•1397 
.1415 




•1547 

Kxsec. 



d. 

1 1 
iT 
II 
12 
iT 
12 

12 
12 
12 
12 
12 

13 
12 
13 
13 
13 
13 
13 

13 
13 
14 
14 
14 
14 

14 
14 
14 
14 
15 
15 

15 
15 
15 
15 
16 

15 
16 
16 
16 

16 
16 

17 

16 

17 
17 
17 
17 
17 
18 

17 
18 
18 
18 
18 
18 
18 
19 
19 
19 
19 

d. 



30 

10 
20 

30 
40 

50 

31 

10 
20 

30 
40 

50 

32 

10 
20 

30 
40 

50 

33 

10 
20 

30 
40 

50 

U 

10 

20 

30 
40 

50 

35 

10 
20 

30 
40 

50 
3(1 

10 
20 

30 
40 

50 

37 

10 
20 

30 
40 

50 

38 
10 
20 

30 
40 

50 

39 

10 

20 

30 
40 

50 

40 



Vers. 



1339 

1354 
1369 

1383 

1398 

1413 

_14^8 

1443 
.1458 
• 1473 
,1489 
.1504 



Kxsec 



1519 



1535 
1550 

1566 

1582 

159^ 



1613 



1629 
1645 
1661 
1677 
1693 



1709 



1726 
1742 

1758 

1775 
1792 



1808 



1825 
1842 
1859 
1876 
1893 



1910 



1927 

1944 
1 961 
1979 
1996 



2013 

.2031 
,2049 
.2066 
.2084 
.2102 



2120 



2138 
2156 
2174 
2192 
.2216 



222g 



.2247 
. 2265 
.2284 
.2302 
.2321 

~233§ 

Vers. 



•1547 



.1566 
.1586 
. 1606 
. 1626 
. 1646 



.166S 



1687 

I/O? 

1728 

1749 
1776 



1792 



• 1813 
.1835 
.1857 
.1879 
. 1 901 



•1923 



.1946 
.1969 
.1992 
.201 5 
• 2038 



.2062 



.2086 

.2110 

.2134 

•2158 
.2183 



.2207 



22 "12 
.2258 
.2283 
.2309 
• 2334 



2366 



.2387 

•2413 
.2440 

.2467 
•2494 



2521 

• 2549 

•2576 
.2604 

• 2633 
.2661 



2690 



.2719 

• 2748 
.2778 

.280^ 

.2837 



.286^ 



.2898 
.2928 

•2959 
.2991 
• 3022 

"3054' 

Kxsec. 



19 
19 
20 
20 
20 
26 

26 
26 
21 
21 
21 
2T 

21 

21 
22 
22 
22 
22 

23 
22 

23 
23 
23 
24 

24 
24 
24 
24 
24 
24 

25 
25 
25 
25 
25 
26 

26 
26 
26 

27 
27 
27 

27 

2f 
28 
28 
28 
28 

29 
29 
29 
29 
30 
30 

30 
30 
31 
31 
31 
31 

d. 



V. V 



31 30 29 28 



3-» 

6.2 


30 
6.0 


2.9 
5.8 


9-3 


9.0 


8.7 


12.4 


12.0 


II. 6 


15-5 
18.6 


15-0 
18.0 


14.5 
17-4 


21.7 


21 .0 


20.3 


24.8 


24.0 


23.2 


27.9 


27.0 


26.1 



5.6 
8.4 



14.0 

16.8 

19.6 
22.4 
25.2 



27 26 25 24 



2.7 


2.6 


2.5 


5-4 
8.1 


5-2 
7.8 


5.0 
7 5 


10.8 


10.4 


10 


135 
16.2 


13.0 
15.6 


12.5 
15 


18.9 


18.2 


'7-5 


21.6 


20.8 


20.0 


24-3 


23-4 


22.5 



24 

4.8 

7.2 

9.6 

T2 .0 
14 4 

16.8 
19.2 
21.6 



23 22 21 20 



2.-, 


2.2 


2.1 


4.6 


4.4 


42 


6.9 


6.6 


6.3 


9.2 


8.8 


8.4 


II. s 


IT.O 


iO.,S 


13-8 


13.2 


13.6 


16. 1 


iS-4 


14.7 


18.4 


17.6 


16.8 


20.7 


19.8 


18.9 



19 18 



1.9 


1.8 


^•7 


3^« 


3^6 


3^4 


5 7 


5-4 


51 


7 6 


7.2 


6.8 


9-5 


9.0 


8.5 


II 4 


I0.8 


10.2 


«3-3 


12.6 


11 .0 


15-2 


14.4 


13-^ 


17.1 


16.2 


15-3 



17 16 

1.6 

3-2 
4.8 



9.6 

II. 2 
12.8 
14.4 



15 14 13 12 



1.5 


1.4 


13 


30 


2.8 


2.6 


4-5 


4.2 


3^9 


6.0 


5-6 


$•2 


7-S 


7.0 


(^■5 


9.0 


8.4 


7.8 


10.5 


9.8 


9.1 


12 .• 


11 .2 


10.4 


13-5 


12.6 


11.7 



2-4 

3^6 

4.8 
6.0 
7.2 

8.4 
9.6 
[0.8 



II 10 O 

1 .1 [i .0 0.0 

2 .2|2 .0 

3-? 30 

I 

o 



4-4 4 

5-5 5 
6.6,6.0 



7-7 7- 
8.8 8. 
9.9I9. 



0.3 

0-3 
0.4 
0.4 



r. V. 



445 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 

40°-50" 50-60" 



' I Vers, j d. Exsec. d. 



40 

lO 
20 

30 
40 
50 

41 

10 

20 

30 
40 

50 

42 

10 

20 

30 
40 

50 

43 

10 

20 

30 
40 

50 

44 

10 

20 

30 
40 

50 

45 

10 

20 

30 
40 

50 

46 

10 
20 

30 

40 

50 

47 

10 

20 

30 

40 

50 

48 

10 

20 

30 
40 

50 

49 

10 
20 

30 
40 

50 

50 



2339 



2358 
2377 
2396 
2415 
2434 



2453 



2472 
2491 
2510 
2529 
2549 



2568 



2588 
26of 
2627 

2647 
266,^ 



2686 



2706 
2726 

274§ 
2765 
2786 



28og 



2827 

2847 

286f 
2888 
2908 



2920 



2949 
2970 
2991 
301 T 
30^2 



3053 



3074 
3095 
3116 
313? 
315^ 



3180 



3201 
3222 

3244 
3265 

3287 



3308 



3330 
3352 
3374 
3395 
34if 



3439 



346T 
3483 
3505 
352? 
3550 



3572 

Vers. 



9 
9 
9 
9 

9 
9 
9 
9 
9 
9 

9 
9 

9 

20 

19 

20 

20 
20 
20 

20 
20 
20 

20 
20 
20 
26 
20 
20 

20 
20 
21 
20 
21 
21 
21 
21 
21 
21 
21 
21 

21 
21 
21 
21 

21 
21 

22 
21 

22 
21 

22 
22 
22 

22 

j 22 
22 
22 



3054 



3086 
3II8 
315I 
3183 
3217 



3250 



3284 
331^ 
3352 
3386 
3421 



345S 



3491 
352^ 
35^3 
3599 
3636 
3673 



3710 

3748 
3786 
3824 
3863 



3901 



3941 
3980 
4020 
4066 
4101 



4142 



41^3 
4225 
4267 
4309 
4352 



439^ 



4439 
4483 
452^ 
4572 
461^ 



4663 



4945 



4993 
5042 
5091 
514T 
5192 



5242 



5294 
534? 
539? 
5450 
5503 



4708 

4755 
4802 

4849 I 4^ 
4896 I 48 



32 

32 
32 
32 
33 
33 
34 
33 
34 
34 
34 
3l 

35 
36 
36 
36 
37 
37 

37 
37 
38 
38 
39 
38 

39 

39 
40 

40 
40 

41 

41 
41 
42 

42 
43 
43 

43 
44 
44 
44 
4? 
4? 

45 
46 
47 
47 



48 
48 
49 
50 
50 
50 

51 
51 
52 
53 
53 
53 
5557 

Exsec. d. 



50 

10 

20 

30 
40 

50 

51 

10 
20 

30 
40 

50 

52 

10 

20 

30 

40 

50 

53 

10 
20 

30 
40 

50 

54 

10 
20 

30 
40 

50 

55 

10 
20 

30 
40 

50 

56 

10 
20 

30 
40 

50 

57 

10 
20 

30 
40 

50 

58 
10 
20 

30 

40 

50 

59 

10 
20 

30 
40 

50 

60 



Vers. 



3572 



3594 
3617 

3639 
3661 

3684 



3707 



3729 
3752 
3775 
379f 
3820 

3843 



3866 
3889 
3912 

3935 
3958 



3982 



4005 
4028 
4052 

4075 
4098 



4122 



4145 
4169 

4193 
4216 
4246 



4264 



4286 
4312 

4336 
4360 

4384 



4408 



4432 

4456 
4486 

4505 
4529 



4553 



4578 
4602 
4627 
465T 

4676 



4701 



4725 

4750 

4775 
4800 

4824 



4849 



4874 
4899 
4924 
4949 
4975 



5000 

Vers. 



22 
22 
22 
22 
22 
23 
22 
22 

23 

22 

23 
23 

23 
23 
23 
23 
23 
23 

23 
23 
23 
23 
23 
23 

23 
24 
23 
23 
24 

23 
24 
24 
24 
24 
24 
24 

24 

24 
24 

24 

24 

24 
24 
24 
24 
24 
24 
25 
24 
24 
25 
25 
24 
25 
25 
25 
25 
25 
25 
2'5 



Exsec. 



d. 



5557 



561 1 
5666 
5721 

5777 
5833 



5890 



594? 
6005 
6064 
6123 
6182 



6242 



6303 

6365 
6427 
6489 
6552 



6618 



6681 
6746 
6811 
6878 
6945 



7013 



708T 
7156 
7220 
7291 

7362 



213£ 



750? 
7581 
7655 
7730 
7806 

7883 

7966 
8039 
8118 
8198 
8279 



8361 



8443 
8527 

8611 

8697 

8783 



8871 



8959 
9048 

9139 
9230 

9322 



0416 



9510 
9606 

9703 
9801 
9900 



I 0000 

Fxsec. 



53 
54 
54 
55 
56 
56 

5? 
58 
58 
59 

59 
60 

61 
61 
62 

62 

63 
64 
64 
65 
65 
66 
67 
68 
68 

69 
70 
76 

71 

72 

73 

73 
74 
75 
75 
77 

71 
7l 

79 
80 
81 
82 

82 

83 
84 

85 
86 



89 
96 

91 
92 

93 
94 
95 
97 
98 

99 
100 

d. 



l». P. 



987654 



0.9 
1.8 
2.7 

3-6 
4-5 
5-4 

6.3 
7.2 



o.8|o.7 
611 .4 
42.1 



5-6 4.9 

6.4 5.6 
7-216.3 



2.4 
3-0 
3-6 

4.2 
4.8 

5-4 



0.5 0.4 
1.00.8 
1.5 1.2 

2.0 1.6 
2.5 2.0 
3.0 2.4 



3-5 
4.0 

4-5 



2.8 
32 
3-6 



3 2 19 8 7 



I 


0.3 


0.2 


0.1 


0.9 


0-8 


2 

3 


o.b 
0.9 


0.4 
0.6 


0.2 
0.3 


1.9 

2-8 


1-7 

2-5 


4 


1 .2 


0.8 


0.4 


3.8 


3-4 


5 
6 


1-5 

i..t 


1 .0 
1.2 


0.5 
0.6 


4-7 
5-7 


4 2 

5-1 


7 
8 


2.1 

2.4 


1.4 
1.6 


0.7 
0.8 


6.6 

7.6 


5-9 
6.8 


9 


2-7 


1.8 


0.9 


«.5 


7-6 



07 
I.. 5 

2.2 
3-0 

3-7 
4-5 

5-2 
6.0 
6.7 



6 5 


4 


3 


2 


I 


o.go.g 


0.4 03 


0.2 


i 


1.3 1 . 1 

1.9 1.6 


0.9 

1-3 


0.7 

1 .6 


0.5 

0.7 


0.3 

0.4 


2.6 2.2 


1.8 


1.4 


1 .0 


0.6 


3.2 2.7 


2.2 


i-7 


1 .2 


0.7 


3-9 3-3 


2-7 


2.1 


1 .5 


0.9 


4-5 3-8 


3-i 


2.4 


1-7 


1 .0 


5.2 4-4 
5-8 4-9 


3 t^ 
4.0 


2.> 

3-1 


2.0 
2.2 


T .2 

1-3 



25 25 24 24 23 23 



2.5 


2.5 


2.4 


2.4 


2-3 


5-1 


S-c 


4-9 


4.!> 


4-7 


7-6 


7-5 


7 3 


7-2 


70 


10. 2 
12.7 


10.0 
12.5 


9 £ 
12.2 


9.6 
12.0 


9.4 
It. 7 


15.3 


150 


14 7 


14.4 


14.1 


^7-8 


17-5 


J7.1 


16.8 


16.4 


20.4 
22.9 


20.0 
22.5 


19. t 
22.6 


10.2 
21.6 


:8.8 
21 .1 



2-3 

4.6 
6.9 

9.2 

lis 

13-8 



4 
20.7 



22 22 21 21 20 20 



4.4 
6.6 



13.21 



4-3 
6.4- 



4 2 

6.3 



8.6 8.4 
[0.710.5 
12.9 12.6 



41 
6.1 



e.o 



8.0 

10. o 

12.31 12 .0 



15.4I15.0 14.7 14-3 M- 
17.6117.21 16.8 I6.4J16 

19.8119.3 18.9ll8.4il8 



19 19 18 



3-9 
5 8 

7.8 

9 7 



13-6 '3 
i5-6|i5 
i7-5i'7 



P. P. 



3-7 
5-5 



7-4 
g.2 



12.9 
14.8 
16.6 



446 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 

G0°-70" 7()°-8()" 



Vers. 



GO 

lo 
20 

30 
40 

50 
Gl 

10 
20 

30 
40 

50 
G2 

10 
20 

30 
40 

50 
G3 

10 

20 

30 

40 

50 
64 

10 

20 

30 

40 

50 
G5 

10 

20 

30 
40 

50 
G6 

10 

20 

30 
40 

50 

67 

10 

20 

30 
40 

50 

68 

10 

20 

30 
40 

50 

69 

10 
20 

30 
40 

50 

70 



5000 



5025 
5050 
5076 
5101 
5126 



5152 



5177 
5203 

5228 

5254 
5279 



530g 

5331 
5356 
53^2 

5408 
5434 



5460 



5486 
5512 

5538 
5564 

5590 



5616 



5642 
5668 

5695 
5721 

574^ 



5774 



5800 

5826 

5853 

5879 
5906 



5932 



5959 
5986 
6012 
6039 
6066 



6092 



61 19 
6i46 
6173 
6200 
6227 



6254 



6281 
6308 

6335 
6362 
6389 



6416 



6443 
6476 

6498 

6525 

6552_ 

6580 

Vers. 



25 

25 
25 
25 
25 
25 
25 
25 
2S 
2S 

25 
26 

25 
25 
26 
26 

2l 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 

26 

26 

26 
26 

26 

26 
26 
26 
26 

26 

27 

26 
26 

27 

26 

27 
27 

26 

27 
27 
27 
27 
27 
27 

27 
27 

2^ 

27 

27 
27 
27 

2f 
d. 



Kxsec. 



,0000 



.0101 
.0204 
.030^ 

•0413 
.0519 



062§ 



.073S 
.0846 
.0957 

. 1076 

,1184 



1300 



I4I8 

i53§ 
1657 

1902 



2027 



2153 
2281 
2411 

2543 
2676 



28ii 



2948 
3087 
3228 
3371 
351? 



3662 



3810 
3961 
4114 
4269 
4426 



4586 



4747 
4912 

5078 
5247 
5419 



5593 



5770 

5949 
613T 
6316 
6504 



6694 



6888 

7085 

.7285 

,7488 

7694 



7904 



811^ 
8334 
8554 
.8778 
9006 



9238 

Kxsec. 



01 
02 
03 

05 
06 
07 
09 
16 
II 

13 
14 
16 

18 
26 
21 
23 
25 

26 

28 
30 
31 

33 
35 

37 

39 
40 

43 
44 
46 
48 
51 
52 
55 
Si 
59 
61 
64 
66 
69 
71 
74 
77 

79 
82 

85 
88 

96 

94 

96 
200 
203 
206 
210 

213 

216 
226 
224 
22^ 
232 

(I. 



70 

10 

20 

30 
40 

50 

71 

10 
20 

30 
40 

50 

72 

10 
20 

30 
40 

50 

73 

10 
20 

30 
40 

50 

74 

10 

20 

30 
40 

50 

75 

10 
20 

30 
40 

50 

76 

10 
20 

30 
40 

50 

77 

10 

20 

30 
40 

50 

78 
10 
20 

30 
40 

50 

79 

10 
20 

30 
40 

50 

80 



Vers. 




6580^ 

6607 
6634 
6662 
6689 

67j7_ 

^744. 

6772 
6799 
6827 
6854 
6882 



6910 



7104 
7132 
7160 
7187 
7215 



7243 



6271 
7299 
732^ 
7355 
7383 



21 



12 



7440 
7468 
7496 

7524 

7552 



7581 



7609 

763? 
7665 
7694 
7722 



775o_ 

7779 
7807 

7835 
7864 
7892 



7921 



7949 
7978 
8005 

8035 
8063 



8092 

8126 
8149 
817^ 
8206 
8235 
8263 

Vers. 



27 
2^ 
27 
2? 
27 
27 

2^ 

2? 
27 
27 
27 
28 

27 

27 
28 
27 
28 
28 

2? 
28 
28 

2^ 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 

2g 
28 

28 
28 
28 

28 

28 

28 

28 
28 

28 

28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 

29 

28 

28 

(1. 



Kxsec. 



9238 

•9473 
•9713 

•995^ 
.0205 

• 0458 

097 1 
.1244 

.1515 
.1792 

.2073 



.2366 

7265^ 
.2951 

• 3255 

• 3565 
.3881 



2.4203 



•4531 
.4867 

.5209 

•5558 
J9i,5 
.6279 



.6651 
.7031 
.7420 

.7816 

. 8222 



8637 



.9061 

.9495 
.9939 
.0394 
.0859 



1335 

1824 
2324 

2836 
3362 
3901 



4454 



5021 
5604 
6202 
6816 
7448 



809f 



8765 
9451 
OI58 
0885 
j[636 

2408 

3205 

4026 

4874 

5749 
6653 

758? 



Kxsec. 



235 
240 
244 
248 

253 
257 
262 
26^ 
276 

276 
281 
287 
292 
298 

304 
310 
316 
322 

328 
33^ 
342 
349 
356 
364 

372 
380 

388 
396 
406 
414 

424 
434 
444 
454 
465 

476 

488 
500 
512 
525 
539 
553 

56? 
582 

598 
614 
631 
649 

667 

686 

707 
728 

749 
772 

796 
821 

847 
^7l 
904 

934 
(I. 



I'. 1* 



9 

0.9 



8 7 



2.7 

3-6 

5-4 

6.3 
7.2 



0.8 
1.6 
2.4 



0.7 
1.4 



2. 1 



3.2 2.8 
4-0 3.5 
4.8 4.2 



5-6 
6.4 
7.2 



4.9 
5.6 



1.8 



2.4 

3-0 
3.6 

4.2 

4.8 
•3 5-4 



5 4 

).S 0.4 
: .0 

-5 



1.2 



2.0 1.6 
2.5 2.0 
302. 4 

3.52.8 
40 3.2 
4.513.6 



3 2 19 8 7 



0-3 


0.2 


0.1 


0.9 


0-8 


0. 


0.6 


0.4 


0.2 


1.9 


1.7 


I. 


0.9 


0.6 


0.3 


2-8 


2.5 


2. 


1 .2 


0.8 


0.4 


3.8 


3-4 


3- 


1-5 


I.O 


0.5 


4-7 


4.2 


3- 


i.fc 


1.2 


0.6 


5-7 


5-1 


4- 


2.1 
2.4 


1.4 
1.6 


0.7 
0.8 


6.6 
7.6 


5-9 
6.8 


5- 
6. 


2.7 


1.8 


0.9 


8-5 


7-6 


6. 



6 5 4 3 2 1 

0.6 0.5 0.4 0.3 0.2 oi 
i.3|i.i 0.9 0.7 0.5 o 3 
i.9|i.6 !•§ 10 0-7 0.4 

2.6|2.2 1.8 1.4 1.00.6 
3.2 2.7 2.21.7 '•2 0.7 
3.9 3.3 2.7|2.1 l.S 0.9 

4.5 3§ 3-1 2.4 1.7 1.0 

5.2 4.4 3.62.^ 2.0 1.2 

5-8 4'94-ol3-i 2.2 r.3 



29 28 28 2f 



2.9 
5.8 
8.7 

II. 6 

14.5 
17.4 

20.3 
23.2 
26. 1 



2-8 

5-7 
8.5 

II. 4 
14.2 
17.1 

19.9 
22.8 
25-6 



2.8 
5.6 
8.4 



14.0 
16.8 

19.6 
22.4 
252 



27 2g 26 2^ 



2.7 


2-6 


2.6 


2-5 


5^4 


.S • 3 


5-2 


51 


8.1 


7-9 


7.8 


7-6 


10.8 


10.6 


10.4 


10.2 










»J 5 


13-2 


13.0 


12.7 


16.2 


'5-9 


15.6 


»5-3 


18.9 


18.5 


18 2 


17-8 


21.6 


21.2 


20.8 


30.4 










24.3 


23-8 


23.4 


22.9 



447 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 

80°-85" 85^-90° 



so 

lO 

20 



40 
50 

81 

10 

20 

30 
40 

50 

S2 

10 

20 

30 
40 

50 

83 

10 

20 

3« 
40 

50 

84 

10 

20 

30 
40 

50 

85 



Vers. 



8263 



8292 
8321 

8349 
8378 
8407 



8435 



8464 

8493 
8522 

8550 
8579 



8608 



8637 

8666 
8694 
8723 
8752 



8781 



8810 
8839 
8868 
8897 
8926 



8954 



8983 
9012 
9041 
9070 
9099 



9128 
Vers. 



<]. 



29 

28 

28 
29 

28 
29 

28 

29 

28 

29 

29 

28 

29 

28 

29 
29 

29 

28 

29 
29 
29 
29 

28 

29 

29 

29 

29 
29 

29 

d. 



Exsec. 



(1. 



4 7587 



4-8554 
4.9553 
5.0588 
5 . 1 666 

5.2772 



5-3924 



5.5121 

5-6363 
5-7654 
5.8998 

6.0396 



6.1853 



6.3372 

6.4957 
6.6613 
6.8344 
7.0I56 



7-2055 



7 • 4046 
7.6138 

7.8336 
8.0651 
8.309T 



8.5667 



8.8391 
9.1275 
9-4334 
9-7585 
10. 1045 



10-4737 

Exsec. 



966 

999 
035 
072 
III 

152 

196 
242 
291 
343 
398 

456 
519 
585 
656 

731 
812 

898 

991 

2091 

2198 

2315 
2440 

2576 

2723 
2884 

3059 
3250 

3466 

3691 



85 

10 

20 

30 
40 

50 

86 
10 
20 

30 
40 

50 

87 

10 

20 

30 
40 

50 

88 
10 
20 

30 
40 

50 

89 
10 
20 

30 

40 

50 

90 



Vers. 



9123 



9157 
9186 
9215 
9244 
9273 



9302 



9331 
9366 

9389 
94I8 

9447 



?47i 

9505 
9534 
9564 

9593 
9622 



9651 



9680 
9709 
9738 
9767 
9796 



982S 



9854 
9883 
9912 
9942 
9971 
0000 

Vers. 



(1. 

29 
29 
29 
29 
29 
29 

29 
29 
29 

29 
29 

29 
29 

29 
29 

29 
29 
29 
29 

29 
29 

29 

29 

29 

29 
29 
29 
29 
29 
29 

d. 



Exsec. 



0-4737 



0.8683 
I .2912 

1-7455 
2.2347 
2.7631 



d. 



3-3356 



3-9579 
4.6368 

5-3804 
6. 1984 
7. 1026 



8.1073 



9.2303 
20.4937 
21 .9256 
23. 5621 
25-4505 



27.6537 



30.2576 

33.3823 
37.2015 

41-9757 
48, 1 146 



56.2987 



67-7573 
84.9456 

113-5930 
170.8883 

342..7752 



00 



Exsec. 



.3946 
.4229 
.4542 
.4892 
.5284 

•5725 
.6223 
-6789 
-7436 
.8180 
.9041 

I . 0047 
I . 1230 
I .2634 

I. 4319 
I .6365 
1.8884 
2.2032 

2.6039 

3.1247 
3.8192 

4.7741 
6. 1383 

8.1846 



d. 



P. P. 



29 29 28 



2.9 
5.9 

8.8 

11.8 
14.7 
17.7 

20-6 
23.6 
26.5 



2.9 
5.8 
8.7 

II. 6 
M-5 
17.4 

20.3 
23.2 
26.1 



5-7 
8-5 



11.4 
14.2 
17.1 

19.9 
22.8 
25.6 



448 



TABLE XL— USEFUL TRIGONOMETRICAL FORMUL.€. 



lo 



II 



sin a = 



I _ tan a / \ 

cosec a |/i + tan""^^ ~ 



— cos 2a 



Vi -\- cot'"^ a 



cos a tan a = Vi — cos"^ <z = 2 sin ^<z cos la 

I -\- cos a 2 tan ^^ 



cot ^a I -|- tan"~ ^<z 

I cot a 



vers ^z cot ^a. 



cos iZ = = 



sec a i/i _|_ cot'^ a Vi + tan^" 



I — vers a = sin a cot ^z = i^i — sin^ a = 2 cos"^ J^ — i 
sin a cot la — 1 = cos^ Ja — sin'-' la —1 — 2 sin'- la. 



tana = 



sin d5 sec a 



cot (Z cos a cosec a Vcosec^ a i 

vers 2a cosec 2a = cot a — 2 cot 2a = sin a sec a 



sin 2^ 



I + cos 2a 



= exsec ^ cot ^a = exsec 2^z cot 2a. 



cot tZ = 



cos a 



sin 2(Z 



_ I -f- cos 2a 



tan a sin a 



I — cos 2a sin 2a 

= V^cosec^ a — I =^ cot la — cosec ^. 

vers a = 1 — cos ^ = sin tz tan ^a —■ 2 sin^ ^tz = cos a 

exsec iz = sec a — i = tan a tan la = vers a sec a. 



exsec (Z. 



sin 4^ 



cos la 
tan 1^ 
cot ^a 



= i/ 



vers ^ _ sin a __ vers ^z cos la 
2 



2 cos 2<^ 



sin a 



a/ I -\- cos rt; _ sin dz _ sin ^ sin la 
2 2 sin la 



vers ^ 



= vers a cosec a = cosec a — cot a = 



tan a 



I + cos a 

= cosec a -{- cot a = 



I 4" sec a' 
tan a 



sin a 



exsec « cosec a — cot ^ 



vers la = i — V|(i -|- cos a). 



12 



exsec ^^ 



1/7 — 



i^|(i -j- cos «) 



— I. 



449 



TABLE XL— USEFUL TRIGONOMETRICAL FORMULA. 



13 


2 tan ^ 


I + tan*^ a 


14 


cos 2a = cos'^ ^ — sin^ « = i — 2 sin^ a = 2 cos''^ a — i 




I — tan^ ^ 


I + tan^ a ' 


15 


2 tan a 


Idn 2tf 

I — tan^ a 


16 


cot'^ ^ — I I — tan^ a 

cot 2^ = i cot a — 1 tan ^ = = , 

2 cot a 2 tan a 


17 


vers 2^ =2 sin''^ a — 1 — cos 2^ = 2 sin <^ cos a tan ^. 


18 


tan 2a 2 tan^ a 2 sin''^ a 


cot a I — tan^^ 1 — 2 sm'^ a 


19 


sin (^ ± <^) = sin ^ cos ^ ± cos ^ sin L 


20 


cos (« ± ^) = cos a cos ^ ^ sin a sin <^. 


21 


sin tz + sin /^ =2 sin ^(^ + <^) cos l{a — <^). 


22 


sin a — sin (5=2 sin ^{a — b) cos |(a + ^)- 


23 


cos ^ -|- cos <5 = 2 cos ^(^ + ^) cos J(^ — /5). 


24 


cos a — cos b = — 2 sin |(^ -|- ^5) sin \{a — b). 




Call the sides of any triangle A^ B^ C, and the opposite angles a, b, and 




c. Call J =: i(^ + ^ + C). 


25 


^ ^ jI jg 

tan l(a - ^) = ^ _^ ^ tan |(^ + /^) = ^ _^ ^ cot Jr. 


26 

27 
28 


C - (^ + B) ^^^ f (" + ^;, = (A BfJ" \f + 'I 
cos ^(rt — ^) ^ 'sin ^(a — b) 


sin i« = |/ ^^ 


cosi« = y ' ^^ \ 


29 




30 




Area = Vs{s - A){s - B){s - C) = ^^"'" ^'" . 
^ '^ '^ '^ 2 sin ^ 




1 



450 



INDEX. 



Abutments for trestlef?, 1G7. 

Accuracy of earthwork computations, 
109. 

Accuracy of tunnel surveying, 189. 

Adjustments of instruments, 303. 

Advantages of tie-plates, 260. 

Allowance for shrinkage of earth- 
work, 113. 

American system of tunnel excava- 
tion, 197. 

Angle-bar (rail-joint) — efficiency, 255. 

Arch culverts, 215. 

Area of culverts — method of compu- 
tation, 204. 

Area of culverts — results based on 
observation, 206. 

Area of the waterway — culverts, 203. 

A. S. C. E. standard rail section, 245. 

Austrian system of tunnel excavat'on, 
197. 

Averaging end areas — for volume of 
earthwork, 79. 

Ballast, 220. 

Ballast— cost, 224. 

Ballast — methods of laying, 223. 

Barometric elevations, 6. 

Belgian system of tunnel excavation, 

197. 
Blasting, 142. 
Blasting — cost, 147. 
Borrow-pits — earthwork, 102. 
Bowls (ties), 241. 

Box CULVERTS, 212. 



Bracing — trestles, 16G. 
Bracing (trestles) — design, 184. 
Bridge-joints (rail), 257. 
Bridge spirals, 4. 
Broken-stone ballast, 221. 
Burnettizing — wooden ties, 234. 

Caps (trestle) — design, 184. 

Cars and horses — use in hauling 
earthwork — cost, 134. 

Cars and locomotives — use in hauling 
earthwork — cost, 136. 

Carts — use in hauling earthwork — 
cost, 130. 

Cattle-guards, 216. 

Cattle-passes, 218. 

Center of gravity of side-hill pactions — 
earthwork, 107. 

Central angle — of a curve, 21. 

Chemical composition of rails, 251. 

Cinders (ballast), 221. 

Circular lead-rails — switches, 278. 

Classification of excavate I material, 
148. 

Compound curves, 37. 

Compound curves — application of tran- 
sition curves, 56. 

Compound sections — earthwork, 07. 

Compulations (approximate) from 
})rofiles — earthwork. 111. 

Computation of j)roducts — earthwork, 
90. 

Computation cf volu^me of earth- 
work, 76. 

451 



452 



INDEX. 



Connecting curve from a curved track 

to the inside, 291. 
Connecting curve from a curved track 

to the outside, 290. 
Connecting curve from a straight 

track, 290. 
Construction of tunnels, 195. 
Contractor's profit — earthwork, 140. 
Corbels — trestles, 1G8. 
Cost of ballast, 224. 
Cost of earthwork, 126. 
Cost of framed timber trestles, 174. 
Cost of metal cross-ties, 240. 
Cost of pile trestles, IGl. 
Cost of rails, 254. 
Cost of ties, 232. 

Cost of treating wooden ties, 236. 
Cost of tunneling, 201. 
Creosoting — wooden ties, 233. 
Cross-country route, 3. 
Crossings, 300. 
Crossing — one straight and one curved 

track, 301. 
Crossing — two curved tracks, 301, 
Crossing — two straight tracks, 300. 
Cross-over between two parallel curved 

tracks — reversed curve, 296. 
Cross-over between two parallel curved 

tracks — straight connecting curve, 

295. 
Cross-over between two parallel 

straight tracks, 293. 
Cross-sectioning — field-work, 10. 
Cross-sectioning — for volume of earth- 
work, 73. 
Cross-sectioning irregular sections — 

earthwork, 100. 
Cross-section method of obtaining con- 
tours, 9. 
Cross-sections — ballast, 222. 
Cross-sections of tunnels, 190. 
Culverts, 202. 
Curvature correction — volume of 

earthwork, 103. 
Curve location by deflections, 23. 
Curve location by middle ordinates, 

27. 



Curve location by offsets from the 

long chor'i, 28. 
Curve location by tangential offsets, 

26. 
Curve location by two transits, 26. 

Deflections for a spiral, ■49. 

Design of culverts — elements, 202. 

Design of nut-locks, 268. 

Design of pile trestles, 161. 

Design of ti3-plates, 261. 

Design of track-bolts, 267. 

Design of tunnels, 190. 

Design of wooden trestles, 174. 

Dimensions of wooden ties, 229. 

Ditch -s, 69. 

Drains — tunnels, 195. 

Drill-holes, position and direction — 

blasting, 145. 
Drilling — blasting, 144. 
Driving spikes, 264. 
Durability of metal ties, 238. 
Durability of wooden ties, 228. 

Early forms of rails, 243. 

Earthwork — cost, 126. 

Earthwork surveys, 72. 

Eccentricity of the center of gravity 
of an earthwork ^rcss-section, 104. 

Economics of treated ties, 236. 

Elements of a 1° curve, 22. 

Elements of a simple curve, 21, 

English system of tunnel excavation, 
197. 

Enlargement of headings — tunnels, 
196. 

Equivalent level sections — eartliwork, 
85. 

Equivalent sections — earthwork, 83. 

Existing track — determination of cur- 
vature, 35. 

Expansion of rails, 249, 

Exploding the charge — blasting, 147, 

Explosive, amount required in blast- 
ing, 146. 

Explosives — blasting, 142, 

Extent of use — metal ties, 238. 



INDEX. 



453 



Extent of use of trestles, 153. 
External distances for a 1° curve, 318. 
External distance — simple curve, 21. 

Factors of safety — design of timber 
trestles, 180. 

Failures of rail-joints, 258. 

Fastenings for metal cross-ties, 240. 

Field-work for locating a spiral, 52. 

Fire protection on trestles, 173. 

Five-level sections — earthwork, 92. 

Floor systems of trestles, 1G7. 

Formation of embankmekts. 111. 

Forming embankments — methods, 115. 

Form of excayatioxs and embank- 
ments, 64. 

Forms of rail sections (standard), 244. 

Formulae for re(|uired area of culverts, 
205. 

Foundations — trestles, 1G5. 

Framed timber trestles — cost, 174. 

Framed trestles, 162. 

Free haul— limit, 124. 

French system of tunnel excavation, 
197. 

Frogs, 272. 

Frog angles — trigonometrical func- 
tions, 321. 

Frog number, 273. 

German system of tunnel excavation, 

197. 
Grade line — change, based on mas3 

diagram, 123. 
Grade of tunnels, 192. 
Gravel (ballast), 221. 
Ground-levers, 276. 
Guard-rails — switches, 277. 
Guard-rails — trestles, 169. 

Hauling earthwork — cost, 130. 

Haul of earthwork — computations, 
116. 

Haul of earthwork — method depend- 
ent on distance, 137. 

Haul of earthwork — profitable limit, 
140. 

Headings — tunnels, 195. 



I-beam bridges, 219. 

Instrumental work of locating curves, 

24. 
Iron-pipe culverts, 209. 
Irregular prisinoid — volume, 94. 
Irregular sections — earthwork, 93. 

Joints of framed trestles, 162. 

Kyanizing — wooden tics, 234. 

Lateral bracing — trestles, 167. 
Length of a simple curve, 20. 
Length of rails, 248. 
Level — adjustments, 309. 
Level sections — earthwork, 81. 
Limitations in location, 34. 
Lining of tunnels, 193. 
Loading— design of timber trestles, 

179. 
Loading earthwork — cost, 128. 
Location surveys, 13. 
Logarithmic sines and tangents of 

small angles— table of, 345. 
Logarithmic sines, cosines, tangents, 

and cotangents — table of, 348. 
Logarithmic versed sines and external 

secants — table of, 393. 
Logarithms of numbers — talb of, 325. 
Long chords for a 1° curve, 318. 
Long chord — simple curve, 21. 
Longitudinal bracing— trestles, 166. 
Longitudinals, 241. 
Loosening earthwork — cost, 127. 

Mathematical DESIGN of switches, 
278. 

Mass curve — area, 121. 

Mass curve — properties, 118. 

Mass diagram, 117. 

Mass diagram — value, 122. 

Metal cross-ties — cost, 240. 

Metal cross-ties — fastenings. 240. 

Metal ties, 238. 

Metal ties — form and dimensions. 239. 

Middle areas — for volume of earth- 
work, 79. 



454 



INDEX. 



Middle ordinate — simple curve, 21. 
Modifications of location — compound 

curves, 40. 
Modifications of location — simple 

curves, 31. 
Mountain route, 3. 
"Mud" ballast, 220. 
Mud-sills— trestle foundations, 166. 
Multiform compound curves, 47. 
Multiple-story construction — trestles, 

163. 

Natural sines, cosines, tangents, and 
cotangents — table of, 439. 

Natural versed sines and external se- 
cants — table of, 444. 

Notes — location surveys, 16. 

Number of a frog— to find, 273. 

Nut-locks, 266. 

Obstacles to location, 29. 
Obstructed curve — curve location, 31. 
Old-rail culverts, 213. 
Open cuts vs. tunnels, 200. 
Ordinates of a spiral, 48. 

"Paper location," 13. 

Pile bents, 155. 

Pile-driving formulae, 159. 

Pile-driving methods, 157. 

Pile foundations for trestles, 165. 

Pile-points and pile-shoes, 160. 

Pile trestles, 155. 

Pile trestles — cost, 161. 

Pipe culverts, 208. 

Pipe culverts — construction, 208. 

Pit cattle-guards, 216. 

Ploughs — use in loosening earth, 127. 

Point of curve, 21. 

Point of curve inaccessible — curve lo- 
cation, 33. 

Point of tangency, 21. 

Point of tangency inaccessible— curve 
location, 30. 

Point-rails of switches — construction, 
275. 



Point-switches, 275. 

Portals (tunnel) — excavation, 199. 

Posts (trestle) — design, 1S2. 

Preliminary surveys, 8. 

Preservation of ties — cost, 236. 

Preservative processe 5 for wood- 
en ties, 232. 

Prismoidal correction (approximate) 
for irregular prismoids, 99. 

Prismoidal correction (true) for ir- 
regular prismoids, 95. 

Prismoids, 72. 

Radii of curves — table, 314. 

Rails, 243. 

Rail expansion, 249. 

Rail-gap at joints — effect, 256. 

Rail-joints, 255. 

Rails — chemical composition, 25L 

Rails — cost, 254. 

Rail testing, 252. 

Rail wear on curves, 253. 

Rail wear on tangents, 252. 

Reconnoissance surveys, 1. 

Renewals of ties — regulations, 231. 

Repairs, etc., of plant for earthwork — 

cost, 139. 
Replacement of a compound curve by 

a curve with spirals, 58. 
Replacement of a simple curve by a 

curve with spirals, 53. 
Requirements for a perfect rail- joint, 

255. 
Requirements — spikes, 263. 
Requirements — track-bolts, 266. 
Roadbed — width, 67. 
Roadways for earthwork — cost, 138. 
Rock ballast, 221. 
Rules for switch-laying, 298. 
Ruling grade, 2. 

Scrapers — use in earthwork — cost, 133. 
Screws and bolts (rail-fastenings), 

264. 
Setting tie-plates — methods, 262. 
Shafts — tunnels, 193. 
Shaft (tunnel) — surveying, 187. 



INDEX. 



455 



Shells and small coal— ballast, 221. 

Shoveling: (hand) of earthwork — cost, 
128. 

Shrinkage of oartliwork, 111. 

Side-hill work^cartliwork, 100. 

Sills (trestle)— design, 184. 

Simple curves, 18. 

Slag (ballast), 221. 

Slide-rule — for earthwork computa- 
tions, 90. 

Slopes — earthwork, G5. 

Slope-stakes — position, 75. 

Sodding — efl'ect on slopes, 70. 

Spacing of ties, 229. 

Span — trestles, 164. 

Specifications for earthwork, 148. 

Specifications for wooden ties, 230. 

Spikes, 203. 

Spikes — driving, 204. 

Spirals — required length, 48. 

Spreading earthwork — cost, 138. 

Stadia method of obtaining topog- 
raphy, 12. 

Standard angle-bars, 259. 

Standard stringer bridges, 219. 

Steam-shoveling — earthwork, 129. 

Stifiness of rails — efl'ect on traction, 
247. 

Stone box culverts, 212. 

Stone foundations for framed trestles, 
166. 

Straight connecting curve from a 
curved main track, 292. 

Straight frog-rails— effect, 280. 

Straight point-rails — effect, 281. 

Strength of timber, 176. 

Strength, required elements — trestles, 
175. 

Stringers for trestles — design, 180. 

Stringers — trestles, 167. 

Stub switches, 273. 

Subchord— length, 19. 

Subgrade— form, 68. 

Superelevation of the outer rail on 
curves — general principles, 43. 

Superelevation of the outer rail on 
curves on trestles, 170. 



Superelevation of th? outer rail on 
curves — practical rules, 45. 

Superintendence of earthwork — cost. 
139. 

Supported joints, 257. 

Surface cattle-guards, 217. 

Surface survey's — tunneling, 185. 

SUKYEYIXG — TUNNELS, 185. 

Suspended joints, 257. 
Switchbacks, 4. 
Switch consteuction, 271. 
Switch-laying — practical rules, 298. 
Switch leads and distances, 321. 
Switch-stands, 276. 



Tamping — blasting, 146. 

Tangent distance — simple curve, 21. 

Tangents for a 1° curve, 318. 

Temperature allowances — rails, 250. 

Terminal pyramids and wedges — 
earthw ork, 65. 

Testing rails, 252. 

Three-level sections — earthwork, 87. 

'• Thrown " of a switch, 279. 

Tie-plates, 260. 

Tie-rods, 276. 

Ties, 226. 

Ties — cost, 232. 

Ties on trestles, 170. 

Tile pipe culverts, 211. 

Timber for framed trestles, 173. 

Timber for pile trestles, 157. 

Timber, strength, 176. 

Topographical maps, use of, 5. 

Tkack-bolts, 266. 

Transit — adjustments, 304. 

Transition curves, 43. 

Transition curves — fundamental prin- 
ciple, 43. 

Transition curves — tables of, 322. 

Trestles, 153. 

Trestles — framed, 162. 

Trestles — pile, 155. 

Trestles — posts — design, 182. 

Trestles — required elements of 
strength, 175. 



456 



INDEX. 



Trestles — sills — design, 184. 

Trestles — stringers — design, 180. 

Trestles vs. embankments, 154. 

Tunnels, 185. 

Tunneling — cost, 201. 

Tunnel spirals, 5. 

Turnout (double) from a straight 

track, 287. 
Turnout from the inner side of a 

curved track — dimensions, 286. 
Turnout from the outer side of a 

curved track — dimensions, 284. 
Turnouts with straight point-rails and 

straight frog-rails — table of, 321. 
Two-level ground — for volume of 

earthwork, 80. 
Two turnouts on the same side, 289. 

Underground surveys, 188. 
Unit chord — simple curves, 19. 
Upright switch-stands, 270. 
Useful trigonometrical formulae — table 
of, 449. 



Valley route, 2. 

Ventilation (tunnel) during construc- 
tion, 199. 

Vertex inaccessible — curve location, 
30. 

Vertex — of a curve, 21. 

Vertical curves, 01. 

Vertical curves— form of curves, 62. 

Vertical curves — requ'r.d length, CI. 

Vulcanizing — wooden ties, 232. 



Waterway required for culverts, 203. 
Wear of rails on curves, 253. 
Wear of rails on tangents, 252. 
Weight of rails, 246. 
Wellhouse process — for preserving 

wooden ties, 235. 
Wheelbarrows — use in hauling earth- 

Avork — cost, 132. 
Wooden box culverts, 212. 
"Wodden" spikes, 266. 
Wooden ties, 227. 



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* Reisig's Guide to Piece Dyeing Svo, 25 00' 

Spencer's Sugar Manufacturer's Handbook . . . .16mo, morocco, 2 00- 
" Handbook for Chemists of Beet Sugar Houses. 

16mo, morocco, 3 OO 

Thurston's Manual of Steam Boilers Svo, 5 00' 

Walke's Lectures on Explosives Svo, 4 00' 

West's American Foundry Practice 12m o, 2 50 

" Moulder's Text-book 12mo, 2 50 

Wiechmann's Sugar Analysis Small Svo, 2 50 

Woodbury's Fire Protection of Mills Svo, 2 50 

MATERIALS OF ENGINEERING. 

Strength — Elasticity — Resistance, Etc. 
{See a^s(? Engineering, p. 8.) 

Baker's Masonry Construction Svo, 5 00 

Beardslee and Kent's Strength of Wrought Iron Svo, 1 50 

Bovey's Strength of Materials Svo, 7 50 

Burr's Elasticity and Resistance of Materials Svo, 5 OO 

10 



$5 


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6 


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10 


GO 


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50 




50 


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25 


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50 


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Byrue's Highway Construction 8vo, 

Church's Mechanics of Engineering — Solids and Fluids 8vo, 

Du Bois's Stresses in Framed Structures Small 4to, 

Johnson's Materials of Construction 8vo, 

Lanza's Applied Mechanics 8vo, 

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Merrill's Stones for Building and Decoration 8vo, 

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Rockwell's Roads and Pavements in France 12mo, 

Spalding's Roads and Pavements 12mo, 

Thurston's Materials of Construction 8vo, 

" Materials of Engineering 3 vols., 8vo, 

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Wood's Resistance of Materials 8vo, 

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Small 8vo, 

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(In the press. ) 

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Rice and Johnson's Differential and Integral Calculus, 

2 vols, in 1, small 8vo, 2 50» 
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Rice aud Jobuson's Differential Calculus Small 8vo, $3 00 

" Abricigmeut of Differential Calculus. 

Small 8vo, 1 50 

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Johnson's Theoretical Mechanics. An Elementary Treatise. 
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Jones's Machine Design. Part I., Kinematics. Svo, 1 50 

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2 50 


2 50 


2 00 


2 00 


6 00 


2 00 


5 00 


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1 50 


1 50 


1 50 


1 50 


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7 50 


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4 00 


1 50 


8 00 


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3 00 


1 00 


7 50 


5 00 



Jones's Macbiue Desigu. Part II., Strength and Proportion of 

Machine Parts 8vo, 

Lanza's Applied Mechanics 8vo, 

MacCord's Kinematics 8vo, 

Merriman's Mechanics of Materials 8vo. 

Metcalfe's Cost of Manufactures 8vo, 

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Kunhardl's Ore Dressing in Europe 8vo, 1 50 

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Alloys Svo, 2 50 

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Boyd's Resources of South Western Virginia Svo, 3 00 

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Brush and Penfield's Determinative Mineralogy. New Ed. Svo, 4 00 

13 



Chester's Catalogue of Minerals 8vo, 

Paper, 

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4to, half morocco, 

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MacCord's Slide Valve Svo, 2 00 

lleyer's Modern Locomotive Construction 4to, 10 00 

Peabody and Miller's Steam-boilers Svo, 4 00 

Peabody's Tables of Saturated Steam Svo, 1 00 

" Thermodynamics of the Steam Engine Svo, 5 00 

" Valve Gears for the Steam Engine Svo, 2 50 

Pray's Twenty Years with the Indicator Large Svo, 2 50 

Pupin and Osterberg's Thermodynamics 12mo, 1 25 

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3 00 


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2 50 



Heagan's Steam and Electric Locomotives 12mo, f 2 00 

Routgeu's Thermodynamics. (Du Bois. ) 8vo, 5 00 

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" Manual of the Steam Engine. Part L, Structure 

and Theory Svo, 6 00 

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Construction, and Operation Svo, 6 00 

2 parts. 10 00 

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12mo, 1 50 

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AYclsbach's Steam Eugine. (Du Bois.) Svo, 5 00 

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Adriance's Laboratory Calculations 12mo, 1 25 

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Fisher's Table of Cubic Yards Cardboard, 25 

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15 



MISCELLANEOUS PUBLICATIONS. 

Alcott's Gems, Sentiment, Language Gilt edges, $5 00 

Davis's Elements of Law 8vo, 2 00 

Emmou's Geological Guide-book of the Rocky Mountains. .8vo, 1 50 

Ferrel's Treatise on the Winds 8vo, 4 00 

Haines's Addresses Delivered before the Am. Ry. Assn. ..12mo, 2 50 

Mott's The Fallacy of the Present Theory of Sound. .Sq. IGmo, 1 00 

Richards's Cost of Living 12mo, 1 00 

Ricketts's Historj^ of Rensselaer Polytechnic Institute 8vo, 3 OO' 

Rotherham's The New Testament Criticall}^ Emphasized. 

12mo, 1 50 
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Large 8vo, 2 00 

Totteu's An Important Question in Metrology 8vo, 2 50 

* Wiley's Yosemite, Alaska, and Yellowstone 4to, 3 00 

HEBREW AND CHALDEE TEXT=BOOKS. 

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Green's Elementary Hebrew Grammar 12mo, 1 25 

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Large mounted chart, 1 25 

Ruddiman's Incompatibilities in Prescriptions 8vo, 2 00 

Steel's Treatise on the Diseases of the Ox 8vo, 6 00 

Treatise on the Diseases of the Dog 8vo, 3 50 

WoodhuU's Military Hygiene , .' 16mo, 1 50 

Worcester's Small Hospitals — Establishment and Maintenance, 
including Atkinson's Suggestions for Hospital Archi- 
tecture... oc o.l2mo, 1 25 

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